At the same time, SPASD set policy regarding the allocation of students to elementary schools.  The method decided upon assigns students to schools based upon rigid geographical boundaries to be redrawn every seven years. 

There are several drawbacks to this system.  Some students will be forced to change schools when the boundaries are redrawn, moving them away from friends and familiar environment.

Also, it is difficult to meet and maintain the student-teacher ratio targets for each elementary school.  Given a snapshot of the distribution of children within the district, it would be difficult to draw a single set of boundaries that meets the ratio targets in all nine schools for all seven grades K-6.  The dilemma is magnified when considering the movement of students into and out of the district after the boundaries are drawn.  Additionally, as students progress through grades, the ratio targets change, making them virtually impossible to meet.

For example, a group of 58 kindergarteners with three teachers achieves a ratio of 19.33, which is within the target range for kindergarteners.  When this group moves on to 1^st grade, it is impossible to meet the 1^st grade target range with a whole number of teachers, since 58/2 = 29 and 58/3 = 19.33.

An alternate system for assigning students to elementary schools was developed by the students in the spring 2004 Operations Research II class at the University of Wisconsin - Stevens Point.  They tested the system on the kindergarten classes in a mini-district consisting of three elementary schools.  The system brought all three schools into the target ranges by adding one teacher.

This study focused on determining whether the system could be as effective for the assignment of sixth graders.  The sixth grade problem is more difficult under the rules of this system, giving less flexibility than the kindergarten case.

The Model

The alternative system is based on the following principles:

These principles predetermine the assignment of most students in the district.  The remaining students represent the flexibility with which the model attempts to meet the student-teacher ratio target ranges. 

We define two types of students.  Fixed students are those who must be assigned to a single school based on the criteria above.  For example, a student who lives within walking distance of one school, a student entering kindergarten with an older sibling already attending a school, or a student who is already attending a school are all classified as fixed students. 

In contrast, starter students are those who, based on the above criteria, may be assigned to one of two or more schools.  For example, students in a family moving into or within the district to a residence that is not within walking distance of any school are starters.  Also, kindergarteners who are the oldest in their family and reside outside the walking distances of any school are starters. 

For each starter student, the model defines two decision variables, one for each of the two schools closest to the student’s residence.  For a given starter, the values of these decision variables indicate to which school the starter will be assigned.  In addition, there are decision variables that allow the number of teachers in each school and grade to increase or decrease.  Consequently, the student-teacher ratios can be affected by shifting starter students and altering the number of teachers.  The resulting model is classified as a mixed-integer nonlinear program (MINLP).

The optimal solution to the model is the combination of decision variable values that minimizes the sum of the following penalties:

The penalties are listed in decreasing order of magnitude.  Therefore, when considering possible solutions, the solver chooses the values that most closely meet the student-teacher ratio targets. 

The Sixth Grade Study

Following the results of the three school kindergarten study, a study of the sixth grade for the entire district was undertaken.  The sixth grade problem is harder, because there is a smaller percentage of starters in the sixth grade than in kindergarten, since most students have already been attending school (in fifth grade) and have not moved.  Fewer starters provides for less flexibility to meet the ratio targets.  The goal of the study was to determine to what extent the student-teacher ratios could better meet the ratio targets under the alternative system, given the reduced flexibility offered by the sixth grade. 

The data used was from the 2004-2005 school year.  The entry dates for each student in the class of 2011 was used to determine if that student had recently moved.  If the student had not recently moved, he/she was classified as fixed.  Of the remaining students, those that lived within one mile of one school were also classified as fixed.  All 54 of those yet unclassified were labeled as starters. 

The resulting problem had 123 decision variables.  It was solved at Argonne National Laboratory using the software MINLP developed by Roger Fletcher and Sven Leyffer.  The results are as shown below.

The allocation of students and teachers using rigid boundaries is as follows:

The results, using the alternate system: 

Note: Bold student teacher ratios represent values that are outside the district target ranges.  Half-teachers represent mixed grade classrooms. 

With rigid district boundaries, only 2 of the district’s student- teacher ratio goals were met.  After assigning the students using the alternate system all but three of the student- teacher ratio goals were met, while reducing the number of teachers by two.  This system represents a more efficient use of district resources. 

Implementation Issues

One draw back of the alternate system is that multiple buses may have to come through a neighborhood, which might lead to confusion for younger students.  This confusion can be minimized if the busses are simply marked either with the school’s name or color. 

Another important issue is students who have special needs or who have parents that prefer their child go to a specific school.  The flexibility of our model allows for requests by parents to be easily accommodated by simply changing the status of a student to fixed if that student must be assigned to a particular school.

In order to accommodate the needs of students (e.g. buying school- specific school supplies) and the needs of the SPASD busing system, the program most likely would need to be solved a few weeks ahead of the first day of school.  This is a concern, since students may move into the district throughout the school year.  To cope with this problem, students can be assigned to the school where the student- teacher ratio needs the most improvement.

Conclusion

The alternative system presented above appears to have excellent potential.  This alternative system has proved to be effective at meeting district-set student-teacher ratio targets while minimizing the distances that students must travel to school.  In order to determine the practicality of this alternative system, a study must be undertaken which evaluates all grade levels within the district simultaneously. 

Moving Forward

In January 2006, we met with SPASD cabinet members to present our alternative system and its results.  These school officials were encouraged by our findings and have provided us with the 2005-2006 school year data that will be used to continue our study.  Currently we are developing a model that can analyze all grade levels concurrently.   

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