Kuipers & Rose 2013 Keeping Students with the Curriculum
Keeping Students with the Curriculum
Using a systems dynamics approach to elementary education.
Jurgen Kuipers and Anika Carolina Rose
Delft University of Technology
jurgen.kuipers@gmail.com, anika.carolina.rose@gmail.com
A study was performed by the authors as an exploration into the systems dynamics
of elementary education. The objective of this study was to analyze which factors
help students keep up with the curriculum best in the context of a systems dy-
namics model. Because of the assumption that schools have very little influence on
the socio-economic factors in their area, only factors which can be influenced by
professionals in education were considered for this study. The long term vision of
this study is to enable professionals in education to see a range of possible results
of changes to the school on keeping students with the curriculum (possibly through
an online tool). To achieve this, a systems dynamics model of a generic elementary
school was built using Vensim. The model was tested under a series of systematically
altered conditions and sensitivity analyses were performed. Difficulties in measuring
psychological factors necessitated assumptions regarding certain factors such as lev-
els of teacher enthusiasm. The major findings of the study, are that teacher training
in an archetypal good school and a combination of policy measures in an archetypal
bad school have the largest influence on students keeping up with the curriculum.
1 Problem Description
This study attempts to address the problem of students falling behind the curricu-
lum of a school. Society has an interest in these students becoming productive
members of society. The importance of keeping children at grade level (called with
the curriculum in this paper) cannot be overemphasized [1, 2]. According to Graves,
increasing the graduation rate by 10% would reduce the number of murders in Ore-
gon by 17 and aggravated assaults by 1,300 per year[1]. Further, a study from
Northeastern University shows the seriousness of the correlation between not com-
pleting high school and being incarcerated. Those who did not complete high school
were 63% more likely to spend time in prison than those who completed college [3].
While these studies deal with completing high school, early intervention is key. As
stated by the American Federation of Teachers: ” waiting rarely works” [4], meaning
children that are behind need to catch up as early as possible. Simply waiting for
students to do this themselves results in these students only falling further behind
and consequently later decreases their chances of completing high school. In this
study, intervention in elementary school is analyzed. Earlier intervention is also
possible but outside the scope of this study.
1.1 Goals and Research Question
The vision of this study is to offer insight into education for educational professionals
with the hope that they can see the impact of changes to their school system (in the
1 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
context of the model) on keeping students with the curriculum. In the future, this
could possibly be accomplished through an online tool. Socio-economical factors are
not included, as in general these cannot be influenced by a school.
The goal of this study is to build a model of a generic elementary school using
Vensim. The structure of this model should allow educational professionals to:
e Input their school specific parameters, such as initial students and teachers.
e See how changes in policy measures such as teacher training or instruction
material affect student learning.
Lastly a long term goal of this this study is to ensure students grow up to be pro-
ductive members of society through education. Therefore the model should help to
show which policies prevent students from falling behind.
This study attempts to answer the research question: Which parameters that we can
influence affect whether students perform with the curriculum or fall behind?
2 Model
2.1 Approach
This study creates a rough, first-look model of elementary schools. System dynamics
was chosen instead of other methods, such as agent based modeling, as this study
does not focus on the individual but instead on the way the system interacts as
a whole. For example, student fractions were used to represent the student body.
Furthermore, teaching levels are used to influence the student fractions, not indi-
vidual teachers. The system focuses on feedback loops; these are explained further
in section 2.4 of this paper. The National Institute for School Leadership in the US
also depicts the feedback relation of Professional Development and Leadership to
student achievement College Readiness [5]. Thus, to represent the system dynamics
of a generic elementary school, a model was constructed in Vensim. The text view
of this model can be found in the appendix.
2.1.1 System Boundary
The scope of this study is limited to a generic elementary school. The scope of this
model is twelve years, allowing for two full cycles of students to move through the
school. The concept of the model is to keep students with the curriculum.
Within this framework it is assumed that any student with the proper help can
keep up with the curriculum. Severely disabled children are outside the scope of
this study. Parental support of students, although affecting student learning, is also
not included. This is assumed to be a societal issue that is not within the scope of
this model. Thus extra teachers dependent on the number of behind and challenged
students are used to compensate for a potential lack of parent support.
2 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
2.2 Factors that Influence Student Learning and Teacher
Development
The factors contributing to student learning were first determine. This study was
limited to the following factors:
e Instruction material [6]
e Teacher training
e Level of overall school improvement support from the administrator [7]
e Teacher enthusiasm
e Student teacher ratio [8]
e Good teachers [9]
In the model, these factors influence student learning by contributing to the variable
effective teaching, which then directly influences student learning. Because student
learning is increased by teachers becoming better teachers - good teachers, and be-
cause teachers become better teachers through training, the factors that influence
the effectiveness of teacher training are considered:
e Amount of teacher training given
e Level of overall school improvement support from the administrator [7]
e Openness of teachers to receive training [8]
e Teacher enthusiasm
2.3. Conceptual Model
Using the influencing factors mentioned above, figure 1 shows a conceptual depiction
of the Vensim model built for this study. The figure shows that the main influence
in the scope of this study on student learning is effective teaching. This in turn is
effected by six parameters, with administrator support as an overarching parameter:
e Administrator support
— Instruction material
— Teacher training
* Level of openness of teachers
— Teacher enthusiasm
— Good teachers
3 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
Feedback
Student
Effective Teaching Learning
Administrator Support
Teachers
Figure 1: Conceptual Model of Vensim Elementary School Model
2.4 Structure
Figure 2 depicts the full Vensim model built to represent a generic elementary school
for this study. A legend explaining the coloring of the model can be found in table 1.
The certainty column of table 1 shows which factors are known versus those where
measuring is not possible or clear.
4 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
Table 1: Legend for Full Vensim Model of a Generic Elementary School
TYPE COLOR VARIABLE CERTAINTY
Initial students Certain: school
Inflow rate specific
Initial + Dropout fraction per year Uncertain:
Inputs Green Initial school reputation school specific
Teacher market Uncertain: area
Outside job opportunities specific
Instruction material per student Certain: school
; Dark Teacher Training specific
Policy Administrator
Measures (italic) Fraction of students passing Uncertain:
— Teacher openness to training TALENTED - school specific
Teacher openness to training UNTAL-
ENTED
Actual student improvement 3
si _ Certain: school
Good teacher ratio specific
Model Orange Ratio of with the curriculum to total ee
KPIs ~ — Effective teaching
Teacher enthusiasm Uncertain
School reputation
Weighting Gray These factors, shown in gray determine Uncertain
factors the amount of influence a parameter has
on either teacher enthusiasm or effective
teaching
5 of 31
Keeping Students with the Curriculum
Kuipers & Rose 2013
PPPON TMA :% omMsTYy
6 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
2.4.1 Feedback Loops
Figure 3 below shows the main feedback loops of the model.
schoalepuatation
Increases teacher
“—
effective teaching
student eaing
pod teacher rato envisioned and cure vache enhusiasm
oa | Shell \
improvement | \
- ‘increases good
‘student Improvement ientors \
increases effective \ Le
teaching
‘student bea
inflow of tots wih los teachorenusasm
improvement vdeo
teacher enthusiasm
reinforces schoo! to cuiculum to tts) teaches leaving
sohoo! reputation
a
Figure 3: Feedback Loops
On the left, the student improvement increases effective teaching loop shows: as
teachers teach better, they sce results in their students, which fulfills them and
makes them more enthusiastic about teaching, which again makes them teach even
better. The loop tends towards stagnation as teachers reach a limit of enthusiasm,
with fully improved students. The student improvement increases good teachers loop
shows the same effect, but adds that as teachers are more enthusiastic, they become
better teachers, which in turn makes them teach better.
On the right, the school reputation increases teacher enthusiasm loop shows that
when the school, as a whole, is doing well then teachers become more enthusiastic
and are more likely to stay at the school, which in turn keeps the school reputation
high. When students are not improving, however, the motivation of teachers also
decreases. Other factors such as the student teacher ratio and outside job opportuni-
ties also impact this. The school reputation reinforces school to have better students
loop shows that when the school as a whole is doing well then higher percentages of
students with the curriculum, enter the school. Socioeconomically advantaged par-
ents, who usually have children who are with the curriculum will keep their students
in these schools. The better students lead to less teacher enthusiasm loop shows that
when students are improving, teachers feel fulfilled, but that when the students are
7 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
already with the curriculum, then teachers do not have as large of an impact on
their learning and hence their enthusiasm is stagnant (albeit at a high level, it no
longer increases).
2.4.2 Subscript Structure
Figure 4 below shows the student subscript structure.
emisioned with and
leaming delay behind faction
difference between
‘envisioned and curent
studentlevel
faction of students passing
the grade
| Stulenis
Toa
student infow completion of — Completed
elementary school ——_
infow rate completion faction
ropoult fraction
per year
schoo! reputation
Figure 4: Subscript Structure
Students are composed of a certain amount from each group of the following groups
(the total of each group is the total students):
e Performance (with, behind)
e State (challenged, normal)
e Grade (grades 1 through 6)
Certain amounts of students enter the school at the beginning of each year (inflow
moment) by the inflow rate, which is an initial input specific to the school. Students
enter with fixed amounts of challenged or normal, as this is a physiological measure.
The amount of students who enter as behind or with, however, depends on the
reputation of the school. This shows that as schools perform worse, children who
are with the curriculum (generally socio-economically advantaged children) will find
other schools to attend. Certain amounts of students move through grades based
on the group performance via students passing the grade which moves students to
the next grade at the end of each year [outflow moment] by the fraction of students
pass. For example, if the fraction of students pass is 0.5 for students behind the
curriculum, fifty percent of students behind the curriculum pass; this accounts for
students who are not quite behind enough to repeat an entire year.
8 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
Levels of students in each state change via difference between envisioned and current
student level. The envisioned amount that should be in certain group [with or
behind] is based on the level of teaching effective teaching. The envisioned amount
is compared to the current amount in each state in the school. This difference, with
a learning delay, is then fed back into the school, moving closer to the envisioned
amount. If students move from behind to with, they leave behind and enter with.
Thus the sum of all the difference between envisioned and current student level
should be zero. Students do not move between challenged or normal (as this a
physiological state). Students drop out of elementary school based on a certain
fraction, given as an initial input variable. Unless otherwise specified, only students
behind dropout. Students can dropout at any point in the year. Certain amounts of
students complete elementary school at the end of each year [outflow moment] by the
completion fraction. The teacher subscript structure follows the student structure
with teachers in the following groups:
e Teacher types (good, average, unmotivated)
e Ability (talented, untalented)
e Career stage (mobile, fixed)
2.5 Specification
In table 2, selected equations from the Vensim model used for this study are ex-
plained. Further information on specification can be found in the appendix.
3 Model Behavior
3.1 Complete School
Because exact school data is confidential, system dynamics modeling is a useful
method to maintain the anonymity of schools. Thus, two archetypal schools were
created to analyze the behavior of a complete school in the model. The values from
these schools were composed to reflect what is deemed a good school (table 3) and
a bad school (table 4).
The model was run for a time span of twelve years to ensure two full cycles of el-
ementary school (one cycle is considered six years) were completed. The variable
Ratio of with the Curriculum to Total is used to demonstrate the behavior of the
schools in the model. Figure 5 depicts the ratio of students who are with the cur-
riculum to total students in each school.
Results observed in figure 5 show the good school with a ratio of approximately 0.8
and the bad school with a ratio of approximately 0.65. The oscillations, particularly
seen in the good school, result from both teacher and student learning delays. The
slight disturbances seen occur from the inflow and outflow of students exactly at the
beginning of each year and at the end.
9 of 31
Kuipers & Rose 2013
Keeping Students with the Curriculum
Table 2: Selected Equations with Explanations
VARIABL
Teacher enthusiasm
EXPLANATION OF
A smoothed (by
takes into account:
instruction material
teacher training
student teacher ratio (only if this is unacceptably
high)
student improvement
e school reputation
Administrator support affects the entire range of en-
thusiasm.
Effective teaching
A fraction that takes into account:
e instruction material
e good teachers
e teacher enthusiasm
Difference between envi-
sioned and current stu-
dent level
As explained in section 2.4, this variable compares the
envisioned amount of students in a certain state (the
envisioned amount is based on the level of effective
teaching) and moves toward the envisioned level with
a learning delay.
Inflow/outflow moment
These moments use pulses to show the inflow and out-
flow (or flow to the next grade level) of students at
the beginning and end of each school year. Because
these happen at the beginning or end of the school
year, these moments cause spikes that can be seen in
other variables, such as the total students completed.
10 of 31
Kuipers & Rose 2013
Keeping Students with the Curriculum
Table 3: Archetypal Good School
TYPE INPUT VARIABLE
Total students
Students Per Year
Percent of students challenged
Percent of students behind the curriculum
Quantitative Student to teacher ratio
Dropout fraction [behind students] (per year)
Percent of students who enter the school [nor-
mal]
Percent of students who enter the school [chal- 5%
lenged]
Extra teachers per student behind or challenged 0.5
School reputation 0.8
Instruction material per student 0.9
Administrator support 0.9 [10]
Qualitative Teacher training 0.7
Teacher openness to training TALENTED 0.9
Teacher openness to training UNTALENTED _ 0.7
Fraction of behind students passing 0.97 [11]
Table 4: Archetypal Bad School
TYPE INPUT VARIABLE
Total students
Students Per Year
Percent of students challenged
Percent of students behind the curriculum
Quantitative Student to teacher ratio
Dropout fraction [behind students] (per year)
Percent of students who enter the school [nor-
mal]
Percent of students who enter the school [chal- 8%
lenged]
Extra teachers per student behind or challenged 0
School reputation 0.6
Instruction material per student 0.5
Administrator support 0.8 [10]
Qualitative | Teacher training 0.7
Teacher openness to training TALENTED 0.5
Teacher openness to training UNTALENTED 0.2
Fraction of behind students passing 0.97 [11]
11 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
Good School Bad School
Ratio of with the curriculumto
total [-]
o 12 3 4 5 6 7 8 9 10 11 12
Time [years]
Figure 5: Ratio of With Students to Total Students in a Good and Bad School
3.2 Single Class of Students Flowing Through the School
For the behavioral analysis of a single class flowing through the school, an archetypal
[good] school, found in table 5, was used. Here one full year of students enter the
school. Their progress is tracked by measuring how many move on to the next year,
how many fall behind and eventually how many complete elementary school.
Table 5: Archetypal Single Class
TYPE INPUT VARIABLE VALUE
Total students 100
Percent of students challenged 5%
Quantitative Percent of students behind the curriculum 20%
~ Student to teacher ratio 25
Dropout fraction [behind students] (per year) 0
Extra teachers per student behind or challenged 0.5
School reputation 0.8
Instruction material per student 0.9
Administrator support 0.9 [10]
Qualitative Teacher training 0.7
Teacher openness to training TALENTED 0.9
Teacher openness to training UNTALENTED 0.7
Fraction of behind students passing 0.97 [11]
The Vensim model was run for a time span of seven years, allowing for one additional
year to measure how many students complete one year behind. A Sankey diagram
is used to illustrate this flow of students, as shown in figure 6.
12 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
Held back a year:
° LAL: fi feD— 4 [students]
Inflow 1st grade:
100.0 [students] Sonpleted en time:
Figure 6: Sankey Diagram Showing a Single Year of Students Flowing Through
Elementary School
Figure 6 shows the year starting with 100 students. These students then move to
the next year. The upward flow of students represents the students one year behind.
As seen in the figure, by the end of the six years of elementary school, 96 students
are with the curriculum, while 4 are behind. Considering that the good archetypal
school included 8% challenged students, of which 3% were behind before entering
the school, this behavior is reasonable. The flows occur at given time intervals due
to the inflow and outflow of students to the next grade level exactly at the beginning
and at the end of each year.
4 Policy Options
To determine which policy measures would be effective in the archetypal schools
from tables 3 and 4, a sensitivity analysis was conducted. The policy measures used
are the Policy Measures found in table 1. This sensitivity analysis was conducted
with the good andbad school types found in tables 3 and 4.
The output variables of the sensitivity analysis were:
e Ratio of students with the curriculum to total students
e Total students completed [with]
e Total students completed [behind]
e School reputation
Using the distributions for the sensitivity inputs found in table 6, the sensitivity
analysis was completed using 200 runs, a twenty year time span, the latin hypercube
method and the noise seed 1234.
13 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
Table 6: Policy Measures and Distribution of Sensitivity Inputs
POLICY MEASURE (SENSITIVITY DISTRIBUTION
INPUT) OF SENSITIVITY
INPUTS
RANDOM_UNIFORM
(0.3,1.0)
Instruction material per student
Teacher Training RANDOM_UNIFORM
(0.2,1.0)
Administrator RANDOM_UNIFORM
(0.4,1.0)
Fraction of students behind passing RANDOM_UNIFORM
(0.2,1.0)
Teacher openness to training TALENTED RANDOM_UNIFORM
(0.4,1.0)
Teacher openness to training UNTALENTED RANDOM_UNIFORM
(0.2,0.7)
Using the settings and distributions mentioned, the spread of the final time of the
sensitivity analysis runs were measured. This represents the condition of the stu-
dents (in terms of with or behind) after the time span of the runs are complete. This
was done for each individual policy measure as well as for all of the policy measures
together, called combination. The final time spreads were then compiled into box
plots. The plots for the variable ratio of with the curriculum to total is depicted
in figure 7. This was used as it best highlights the state of school after the policy
measures have been implemented. These plots are specific to the archetypal schools
described in tables 3 and 4.
Good School Bad School
Figure 7: Box Plots of the Variable Ratio of with the curriculum to total for End
Values of Sensitivity Analysis of Policy Measures
In the good school, it can be seen that teacher training has the biggest impact on
increasing the ratio of with students, even greater than if all policy measures are
enacted. In the bad school, the combination of factors is much more effective. The
14 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
arily lead to the best results;
large spread, however, shows that it may not necess
thus, the exact combinations that lead to better results versus worse results need
to be investigated further. Unlike the good school, the bad school seems to be less
affected by teacher training, which could be a result of too many factors holding
back teachers from enacting change.
5 Conclusions
To conclude, the original research question is considered. Which parameters that
we can influence effect whether students perform with the curriculum or fall behind?
Table 7 lists recommendations to accomplish specific goals for both good and bad
schools. These recommendations are drawn from the sensitivity analysis conducted
in section 4. The recommendations show repetitiveness as they are drawn from one
type of archetypal school. Thus, the importance of school specificity is highlighted.
Table 7: Policy Recommendations for Good and Bad Schools
GOA RECOMMENDATIONS
Reduce the number of students that Good: More supportive administrator
are behind the curriculum when and increase of teacher training
completing elementary school Bad: More supportive administrator,
increase of teacher training, and a
stricter pass rate for students that are
behind
Increase number of students who are Good: Increase teacher training and
with the curriculum when completing make sure the administrator stays sup-
elementary school portive
Bad: More supportive administrator,
increase teacher training and instruc-
tion materials
Reaching a higher ratio of with to Good: Increase teacher training and
behind students make sure the administrator stays
portive
Bad: More supportive administrator,
increase teacher training and instruc-
tion materials
Good: Increase teacher training and
make sure the administrator stays sup-
portive
Bad: Make sure that the administrator
stays supportive, then invest in instruc-
tion material
sup-
Improving school reputation
15 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
5.1 Future Work
Uncertainty analysis on the weighting factors should be conducted in future work.
This would determine, for example, the weight of the influence of good teachers.
Using this analysis, it could be determined which factors are more pertinent. Next,
a scenario analysis on which combinations of factors lead to desired and undesired
scenarios should be conducted. This analysis would highlight which combinations
should be avoided most, as well as which combinations are the most desirable. This
specifically applies to combination of policy measures analyzed in section 4.
As mentioned in section 1.1, the vision of this model is to create an online tool
that administrators or other educational professionals can use to help them see how
changes could affect their school. In future work, this system would be created.
Additionally, future work of this study includes incorporating new research by the
Bill and Melinda Gates Foundation on measuring effective teaching into the structure
of the model [12].
References
[1] B Graves. Prisons don’t use reading scores to predict future inmate populations.
The Oregonian, March 2010.
[2] G Romano. Cant read? lets build you a prison cell. Raising a Reader, May
2012.
[3] A Sum, I Khatiwadad, and J McLaughlin. The consequences of dropping out
of high school, October 2009.
[4] American Federation of Teachers. Waiting rarely works: late bloomers usually
just wilt. Readingrockets.org, November 2012.
[5] National Institute for School Leadership. Profes
leaders, 2012.
onal development for school
[6] M Chingos and G Whitehurst. Choosing blindly: Instructional materials,
teacher effectiveness, and the common core, 2012.
[7] G Rose. System dynamics and education/school systems in Pennsylvania [In-
terview with Jurgen Kuipers and Anika Rose], 2012.
[8] D Fisher. Sd on/at schools [Email Corespondence with Jurgen Kuipers and
Anika Rose], 2012.
[9] B Birch. Study: Good teachers have profound effect on students. Education
News, January 2012.
[10] National Center for Education Statistics. Schools and staffing survey, 2008.
{11] J Cannon and $ Lipscomb. Early grade retention and student success, 2011.
16 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
[12] Bill & Melinda Gates Foundation. Measures of effective teaching project releases
final research report, 2013.
Appendix - Vensim Model Text View
following a class 1 year behind[grades 1 to 5,performance|=
SUM(Students[grades 1 to 5, performance, state!]) *
following a class time[next grades 1 to 5\
;performance] ~~|
following a class 1 year behind[g6, performance|]=
SUM(Students[g6, performance ,state!]) * PULSE(TIME STEP
+6,1)
student
: |
following a class 1 year total=
SUM( following a class 1 year[performance!])
~ student
following a class with curriculum [grades , performance|=
(SUM(Students[grades, performance, state!]) «following a
class time [grades , performance \
})
student
following a class 1 year[performance]=
SUM(following a class 1 year behind[ grades! , performance
~ student
: |
teacher enthusiasm=
MIN (MAX(SMOOTH3((instruction material ratio * influence
of instruction material on teacher enthusiasm\
+ teacher training * influence of teacher training on
teacher enthusiasm
+ MIN((desired student teacher ratio/student teacher
ratio)*7,1) * influence of student teacher ratio on
teacher enthusiasm
+ scaled student improvement * influence of student
improvement on teacher enthusiasm
+ school reputation * influence of school reputation on
teacher enthusiasm)
/ (0.8*(influence of instruction material on teacher
enthusiasm + influence of teacher training on teacher
enthusiasm \
17 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
influence of student improvement on teacher enthusiasm
+ influence of school reputation on teacher
enthusiasm \
)) * MAX(MIN
(administrator ,1) ,0)
teacher adaption delay) ,0) ,1)
~ Dmnl
smoothed to prevent the spikes at the end of the
academic year
|
fraction of students passing [| With]=
i]
fraction of students passing [| behind]=
0.97
~ Dmnl
~ Good School 1;0.97;
Bad School 1,0.97;
|
total training effectiveness[teacher type, talented, career
stage|=
teacher openness to training TALENTED * teacher training
«x MAX(MIN(administrator ,1) ,\
0) "|
total training effectiveness[teacher type, untalented,
career stage|=
teacher openness to training UNTALENTED * teacher
training * MAX(MIN(administrator ,1\
) 0)
~ Dmnl
: |
teacher openness to training UNTALENTED=
0.7
~ Dmnl
Good School 0.7
Bad School 0.2
|
instruction material effectiveness= WITH LOOKUP (
teacher training xMAX(MIN( administrator ,1) ,0),
({(0 ,0) —(5,1)},(0 ,0.3) ,(0.1,0.35) ,(0.189602,0.5)
(0.348624 ,0.7) ,(0.648318 ,0.9) ,(1,1\
) (5,1) ))
. Dmnl
instruction material will only be effective if it
is used with training. \
Hence, this lookup shows that after some training (a
18 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
certain threshold \
must be met) training makes the instruction
materials really useful.
difference between envisioned and current student level [
grades , performance , state |=
DELAY3I((envisioned with and behind fraction [performance
| * SUM(Students [grades , performance \
!,state]) — Students[grades , performance ,state]) /
survey period of a year, learning delay\
initial learning difference)
student /year
: |
students passing the grade[grades , performance ,state] :EXCEPT
[g6, performance , state\
Students [grades , performance, state]*fraction of students
passing [ performance ]* outflow moment\
/TIME STEP
student /year
following a class with total=
SUM( following a class with[ performance !])
~ student
following a class time[gl, performance]=
PULSE(TIME STEP, 1 ) ~~|
following a class time[g2,performance]=
PULSE(TIME STEP+1, 1) ~~|
following a class time[g3,performance]=
PULSE(TIME STEP+2, 1) ~~|
following a class time[g4, performance]=
PULSE(TIME STEP+3, 1) ~~|
following a class time[g5, performance]=
PULSE(TIME STEP+4, 1) ~~|
following a class time[g6,performance]=
PULSE(TIME STEP+5,1)
~ Dmnl
° |
following a class with[performance]=
SUM( following a class with curriculum[{ grades! ,
performance |)
student
: |
next grades 1 to 5:
g2, g3, g4, 25, g6 — grades 1 to
ow
19 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
: |
grades 1 to 5:
gl, 82, g3, g4, g5
teacher type:
good, average, unmotivated
student inflow[grades , With, state]=
inflow rate[grades ,state] * school reputation * inflow
moment /TIME STEP ~~|
student inflow [grades , behind, state]=
inflow rate[grades ,state] * (l—school reputation) *
inflow moment /TIME STEP
student /year
° |
scaled student improvement= WITH LOOKUP (
actual student improvement ,
({(-1,0) —(1,1)],(—1,0) ,(—0.1,0) ,(-0.06 ,0.1)
.(—0.02,0.3) ,(0,0.5) ,(0.02 ,0.7) ,(0.06 ,0.9\
) (01,1) ,(1,1) ))
~ Dmnl
initial learning difference=
student /year
reputation delay=
2
~ year
teacher adaption delay=
0.5
~ year
survey period of a year=
1
~ year
influence of instruction material on teacher enthusiasm=
0.2
~ Dmnl
influence of school reputation on teacher enthusiasm=
20 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
0.2
~ Dmnl
influence of student improvement on teacher enthusiasm=
0.5
~ Dmnl
: |
influence of teacher enthusiasm on effective teaching=
0.4
~ Dmnl
. |
influence of teacher training on teacher enthusiasm=
0.45
~ Dmnl
. |
influence of good teachers on effective teaching=
0.5
. Dmnl
: |
influence of instruction material on effective teaching=
0.2
. Dmnl
influence of student teacher ratio on teacher enthusiasm=
0.2
. Dmnl
total students passing the grade[gl]=
SUM(students passing the grade[gl,performance! ,state!]) *
TIME STEP ~~|
total students passing the grade[grades 2 to 5|=
SUM(students passing the grade[grades 2 to 5,performance
!,state!]) *TIME STEP
student
total students completed [performance]=
SUM( Total Completed [ performance, state!])
~ student
° |
envisioned fraction of unmotivated teachers|teacher type,
ability ,career stage]= WITH LOOKUP\
(
teacher enthusiasm ,
({(0 ,0) —(1,1)},(0,0.8) ,(0.25,0.5) ,(0.5 ,0.3)
,(0.75 ,0.13) ,(1,0) ))
~ Dmnl
21 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
average and unmotivated:
average, unmotivated
good and average:
good, average
. |
good teacher ratio=
SUM( teachers [good, ability!,career stage!]) /SUM(teachers [
teacher type!,ability!,career stage\
1)
Dmnl
: |
completion of elementary school[performance, state|=
Students [g6, performance ,state]*completion fraction [
performance ]* outflow moment/TIME STEP
student /year
difference between envisioned and current teacher level [good
ability , career stage]=
DELAY3((envisioned fraction of good teachers[good,
ability ,career stage] * SUM(teachers\
[teacher type!, ability , career stage]) — teachers [
good, ability , career stage]) / \
survey period of a year, teacher development delay)
|
difference between envisioned and current teacher level [
average ,ability ,career stage]\
DELAY3(((1 — envisioned fraction of unmotivated teachers
[average ,ability ,career stage\
]) * (1 — envisioned fraction of good teachers [
average ,ability ,career stage]) * SUM\
(teachers[teacher type!, ability , career stage]) —
teachers[average, ability , career stage\
]) / survey period of a year, teacher development
delay) ~~|
difference between envisioned and current teacher level [
unmotivated , ability ,career stage\
DELAY3((envisioned fraction of unmotivated teachers [
unmotivated, ability ,career stage\
|] * (1 — envisioned fraction of good teachers [
average ,ability ,career stage]) * SUM(\
22 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
teachers[teacher type!, ability , career stage]) —
teachers{unmotivated, ability , career stage\
]) / survey period of a year, teacher development
delay )
teacher /year
(eq 1) : good <> avg + unmotiv teachers via
training
(eq 2 and 3) : unmotiv <> avg teacher via teacher
enthusiasm
|
envisioned fraction of good teachers[teacher type, ability ,
career stage]= WITH LOOKUP \
(
total training effectiveness[teacher type,ability , career
stage],
([(0 0) —(1,1)] .(
,(0.8 0.65) ,(
. Dmnl
0.2,0.08) ,(0.4,0.2) ,(0.6,0.4)
)
Cumulative teachers that have left [teacher type, ability ,
career stage|= INTEG (
teachers leaving|[teacher type,ability ,career stage],
9)
teacher
teacher enthusiasm effect on leaving= WITH LOOKUP (
teacher enthusiasm ,
({[(0 ,0) —(5,1)],(0 ,0.5) ,(0.1,0.3) ,(0.2 ,0.22)
,(0.4,0.1) ,(0.6,0.04) ,(0.8,0) ,(3,0) ))
~ Dmnl
teachers leaving[teacher type, ability , fixed]=
0 ~*|
teachers leaving[unmotivated, ability , mobile]=
teacher enthusiasm effect on leaving * teachers [
unmotivated, ability , mobile] * outflow moment\
/ TIME STEP ~~|
teachers leaving[average, ability , mobile]=
(MAX(teacher enthusiasm effect on leaving, 0)) *
teachers[average, ability , mobile] \
* outflow moment/TIME STEP ~~|
teachers leaving[good, talented, mobile]=
MIN(MAX(teacher enthusiasm effect on leaving + outside
job opportunties, 0),1) * teachers\
[good, talented, mobile] * outflow moment/TIME STEP
~~]
23 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
teachers leaving[{good, untalented, mobile]=
(MAX(teacher enthusiasm effect on leaving, 0)) *
teachers[good, untalented, mobile] \
* outflow moment /TIME STEP
teacher /year
(eq 1) all fixed teachers stay
a frac related to the teacher improvement, teacher
enthusiasm and job \
opportunities elsewhere
|
envisioned percentage of students with the curriculum= WITH
LOOKUP (
effective teaching ,
({(0,0) —(1,1)] ,(0 ,0.3) ,(0.1,0.45
,(0.5,0.65) ,(0.7 9 .
)))
~ Dmnl
° |
necessary instruction material per student=
1
~ Dmnl
teacher development delay=
1
~ year
° |
teacher openness to training TALENTED=
0.9
~ Dmnl
Good School 0.9
Bad School 0.5
|
instruction material ratio=
instruction material per student/necessary instruction
material per student
~ Dmnl
: |
effective teaching=
MAX(0,MIN(1,(instruction material ratio * instruction
material effectiveness * influence of instruction
material on effective teaching \
+ good teacher ratio * influence of good teachers on
effective teaching
+ teacher enthusiasm * influence of teacher enthusiasm
on effective teaching)
/ (0.8*(influence of instruction material on effective
24 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
teaching + influence of good teachers on effective
teaching \
+ influence of teacher enthusiasm on effective
teaching ))))
. Dmnl
total training effectiveness is already treated in
the improvement of \
average teacher —> good teachers. good teacher ratio
is already effected \
by the administrator through the training , however,
how effective these \
good teachers can be is still effected by the
administrator , hence the \
effect is mult. again here.
|
total teachers=
IF THEN ELSE(SUM(teachers[teacher type!,ability!,career
stage !]) >0, SUM(teachers [teacher type\
!,ability!,career stage!]), 1)
teacher
teachers[teacher type,ability ,career stage|= INTEG (
hiring teachers[teacher type,ability ,career stage] +
difference between envisioned and current teacher
level\
[teacher type,ability ,career stage] — teachers
leaving [teacher type,ability ,career stage\
>
initial teachers[teacher type, ability , career stage
))
teacher
: |
ability:
talented , untalented
: |
actual student improvement=
SUM( difference between envisioned and current student
level [grades!,With,state!])/total students\
xsurvey period of a year
~ Dmnl
: |
career stage:
mobile, fixed
25 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
outside job opportunties=
0.05
~ Dmnl
~ 0 to l
0 — no outside opportunities
1 —> very many outside opportunities
teacher gap=
—total teachers+desired teachers
~ teacher
: |
fraction teachers leaving=
SUM(teachers leaving[teacher type!, ability!, career
stage!]) / total teachers * survey period of a year
~ Dmnl
: |
hiring teachers[teacher type, ability , mobile]=
inflow moment/TIME STEP#MAX(0, teacher gap)*teacher
market[teacher type, ability] ~~|
hiring teachers[teacher type, ability , fixed]=
0
~ teacher/year
school reputation=
DELAY3I(MIN(0.3 + 0.8*(ratio of with the curriculum to
total — 0.2*MAX(0,(1—2«fraction teachers leaving \
))),1),reputation delay ,initial school reputation)
Dmnl
base school reputation is set at .3
initial teachers[teacher type, ability , fixed]=
0.2xteacher market[{teacher type, ability]* total students
/desired student teacher ratio\
|
initial teachers[teacher type, ability , mobile]=
0.8*xteacher market{teacher type, ability]* total students
/desired student teacher ratio
teacher
assume only 20% of teachers are intitially ” fixed”
|
teacher market[teacher type, ability]=
0.29 0.01;
0.4 ,0.2;
0.01 ,0.09;
~ Dmnl
26 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
~ Good School 0.29 ,0.01;0.40 ,0.20;0.01 ,0.09;
Bad School 0.14 ,0.01;0.35 ,0.35;0.01 ,0.14;
|
extra teachers per student behind or challenged=
0.05
teacher /student
~ Good School 0.05;
Bad School 0.00;
|
initial school reputation=
0.8
~ Dmnl
~ Good School 0.8
Bad School 0.5
|
previous grades 2 to 5:
gl,g2,g3,g4 —> grades 2 to 5
7 |
inflow moment=
PULSE TRAIN(TIME STEP, TIME STEP, 1, FINAL TIME)
. Dmnl
~ 12 yr PULSE TRAIN(TIME STEP, TIME STEP, 1, FINAL
TIME)
1 yr PULSE(TIME STEP, TIME STEP)
|
grades 2 to 5
g2, 83, g4, 85
inflow rate[grades ,normal]=
92,0,0,0,0,0 ~~|
inflow rate[grades , challenged|=
8,0,0,0,0,0
~ student
~ Good School [normal] 92,0,0,0,0,0; [challenged]
8,0,0,0,0,0;
Bad School [normal] 92,0,0,0,0,0; [challenged]
8 ,0.,0',0 ,0.,0;
|
initial Students[grades , performance , normal]=
85,10;
85,10;
85,10;
85,10;
85.10;
27 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
85,10; ~~|
initial Students[grades , performance , challenged|=
student
Good School 85,10; 3,2;
Bad School 64,28; 1,7;
|
completion fraction [ performance]=
1,0.97
~ Dmnl
~ Good School 1,0.97;
Bad School 1,0.97
|
Cumulative Dropouts[performance ,state]= INTEG (
SUM( dropout [ grades! , performance, state]) ,
0)
student
grades:
gl,g2,293,¢4,25,26
: |
outflow moment=
PULSE TRAIN(0, TIME STEP, 1 , FINAL TIME)
~ Dmnl
. |
dropout fraction per year[ grades , performance]=
0,0;
0,0;
0,05
0,0;
0,0;
0,0;
~ 1/year/year
~ Good School 0,0:
Bad School 0,0.03;
Students[grades 2 to 5,performance ,state]= INTEG (
students passing the grade[previous grades 2 to 5,
performance , state] + student inflow \
[grades 2 to 5,performance ,state] — dropout[grades 2
28 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
to 5,performance ,state] — students passing the
grade\
[grades 2 to 5,performance ,state] + difference
between envisioned and current student level\
[grades 2 to 5,performance ,state],
initial Students[grades 2 to 5,performance, state ])
|
Students [gl , performance ,state]= INTEG (
student inflow[gl,performance ,state] — dropout[gl,
performance ,state] — students passing the grade\
[gl, performance ,state] + difference between
envisioned and current student level[gl\
; performance , state],
initial Students [gl, performance , state ])
Students [g6, performance ,state]= INTEG (
students passing the grade[g5,performance ,state] +
student inflow[g6, performance , state\
] — dropout [g6, performance , state] — completion of
elementary school[performance , state\
] + difference between envisioned and current
student level[g6,performance , state],
initial Students [g6, performance , state ])
student
dropout [grades , performance , state]=
dropout fraction per year[ grades , performance |* Students [
grades , performance , state ]*TIME STEP
student /year
Total Completed | performance ,state|= INTEG (
completion of elementary school[performance , state],
0)
student
. |
learning delay=
0.1
year
envisioned with and behind fraction [With]=
envisioned percentage of students with the curriculum
envisioned with and behind fraction [behind]=
l-envisioned percentage of students with the curriculum
~ Dmnl
performance:
29 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
With, behind
: |
state:
normal, challenged
: |
total students=
IF THEN ELSE(SUM(Students[grades!, performance!, state !])
>0, SUM( Students [ grades! , performance \
!,state!]), 1)
student
: |
ratio of with the curriculum to total=
SUM(Students[grades!, With, state!])/total students
~ Dmnl
: |
student teacher ratio=
total students/total teachers
~ student / teacher
desired teachers=
SMOOTH3(total students/desired student teacher ratio + (
SUM(Students[grades!, behind\
, state!]) + SUM(Students[grades!, performance! ,
challenged])) * extra teachers per student behind
or challenged\
; teacher adaption delay)
teacher
smoothed to prevent spikes in delay during end of
academic year
|
administrator=
0.9
~ Dmnl
Good School 0.9
Bad School 0.8
desired student teacher ratio=
25
~ student /teacher
Bood School 25;
Bad School 30;
instruction material per student=
0.9
30 of 31
Kuipers & Rose 2013 Keeping Students with the Curriculum
. Dmnl
Good School 0.9
Bad School 0.5
|
teacher training=
0.7
~ Dmnl
~ Good School 0.7
Bad School 0.7
|
2 2 2 2 2 2 2g 2 2 a go oo 2K 2 2 OO OK OK OK oo 2 2 oo Ko KK ok OK KK KOK KK OK KK KKK
. Control
FESO ISI GIGI III GIS II a”
Simulation Control Parameters
|
FINAL TIME = 12
~ year
~ The final time for the simulation.
|
INITIAL TIME = 0
~ year
. The initial time for the simulation.
|
SAVEPER =
TIME STEP
year [0,7]
The frequency with which output is stored.
|
TIME STEP = 0.03125
~ year [0,7]
The time step for the simulation.
31 of 31
