Curriculum Optimization Project

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Curriculum Optimization Project

• CURRICULUM REFORM MEETING
• California State University, Los Angeles
• Kevin Byrnes
• Dept. of Applied Math
• Johns Hopkins University

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Project Goals

• 1) Topics grouped into courses should be closely
  related
• 2) The curriculum should allow flexibility
• 3) The new curriculum should be completable in
  the same time needed for the current one.

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Model of the Problem as a Graph

• Topics correspond to vertices (nodes)
• Directed edges correspond to a direct
  prerequisite relationship between topics

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A Shortest Path

• The length of a shortest (undirected) path
  between two vertices A and B in a graph is the
  minimum number of edges one must traverse in the
  graph to get from vertex A to vertex B.

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From Topics to Courses

• One idea Identify seeds for a fixed number of
  courses, then try to assign each topic to a
  seed/course.
• We can estimate the number of seeds we want by
  the number of courses currently in the curriculum.

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How to identify seeds

• The most important topics are those which have
  edges coming from and going to many other
  courses. Why not identify the N vertices in our
  graph with the largest number of incoming and
  outgoing edges?
• (Idea due to Kleinberg)

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Goals in Constructing a Course

• We would like to assign each topic to its nearest
  seed (course) for intellectual cohesiveness (goal
  1)
• Well have some constraints, however
• 1) Every topic should be assigned to a course.
• 2) The resultant number of credit hours for a
  course should be bounded.

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Variables for IP1

• xij 1 if topic I is assigned to course j, 0
  otherwise.
• ci the estimated number of credit hours needed
  to teach topic I
• tij the length of the shortest undirected path
  in G from topic I to seed j
• lb lower bound on credit hours for a course
• ub upper bound on credit hours for a course

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Explanation of Constraints

• 1) and 3) together guarantee that each topic is
  assigned to a course
• 2) bounds the number of credit hours for any
  prospective course
• OF1 attempts to assign each vertex to its closest
  seed (aka assign each topic to its nearest
  course)

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Interpreting the Output

• The output from IP1 is a doubly-indexed vector
  xij (aka. a matrix). We can interpret this
  output as a set of courses S1,,SN as follows
• Sj the set of xij for which xij 1

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There BackCycling and Other Dangers
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Determining Prerequisity

• Simple Method If topic Sag in course Sa is a
  prerequisite for topic Sbh in course Sb, the
  course Sa is a prerequisite for course Sb

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Some Difficulties

• We may encounter some difficulties with the
  Simple method, such as having one relatively
  small topic as a prerequisite forcing a student
  to take an entire prerequisite course. This
  could result in many prerequisite courses with
  little or no relation to one another.

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An Alternate Method

• Test for Prerequisity For two courses Sa and
  Sb, sum all the credit hours of distinct
  prerequisite topics in Sa, and distinct topics in
  Sb having topics in Sa as prerequisites. If both
  of these sums exceed some threshold values, then
  Sa is a prerequisite for Sb.

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Another Pitfall Directed Cycles

• A directed cycle in a graph is a group of
  vertices, say A, B, and C, such that there is a
  directed edge from A to B, a directed edge from B
  to C, and a directed edge from C to A
• One such example would be three courses that are
  mutual prerequisites

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The Feasibility Condition

• For any course Sb, having course Sa as a
  prerequisite, Sb must be assigned to be taught
  during a semester no earlier than Sa

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Identifying Core Courses

• Having our set of courses, S1,,SN from IP1, we
  would like to identify which of these constitute
  the core of our curriculum. There are several
  ways to do this.
• 1) Have a panel of experts examine the output of
  IP1 and look for natural candidates.
• 2) Create an essentiality index for courses.

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Putting It All TogetherGenerating a Curriculum
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Goals for the Curriculum

• We wish to attain goals 2 3 of the
  introduction. That is, we wish to create a
  flexible curriculum that can be completed in the
  same amount of time as the one in place. We also
  face some constraints
• 1) Each core course must be taken
• 2) The number of courses assigned per semester
  must be bounded
• 3) We want to avoid assigning mutual
  prerequisites in different semesters

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How to Construct the Curriculum

• Well create the curriculum from the courses by
  assigning each required course to a semester to
  be taught. Then, we shall penalize any
  curriculum that assigns mutual prerequisites to
  different semesters

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IP2 variables

• xij 1 if course Si is assigned to semester j, 0
  otherwise.
• ai the expected number of credit hours it will
  take to teach course Si
• Xij denotes the decision variable for a required
  course
• lj and uj are the lower and upper bounds for the
  amount of credit hours to be assigned during
  semester j.
• Epsilon is the upper bound on the number of
  courses to be offered during a single semester.

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The Penalty

• v the measure of how much a proposed curriculum
  violates the Feasibility Condition. If a
  proposed curriculum violates this condition
  noticeably, v should be large and positive
• Alpha is a positive constant

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Explanation of Constraints

• 1) defines dj as the sum of credit hours assigned
  to courses assigned to semester j
• 2) and 6) state that all required courses must be
  assigned to a semester, while 3) and 6) state
  that every elective is assigned to at most one
  semester
• 4) bounds the number of credit hours assigned to
  a semester above and below
• 5) bounds the number of courses assigned to a
  semester
• OF2 minimizes the max over the number of credit
  hours assigned to each semester, and the penalty
  term

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Output From IP2

• IP2 gives us an optimal scheduling of courses for
  a specified value of alpha, epsilon, lj, and uj.
  Letting alpha go to infinity, we can strictly
  enforce the Feasibility Condition as a necessary
  one, but setting it as a penalty enables us to
  initialize IP2, and examine unsatisfactory
  prerequisite relationships in the optimal output

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Where Do We Go From Here?
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The Next Stage

• Implement the methods outlined and develop the
  first curriculum to evaluate them
• Develop a computer program or efficient math code
  (ie. Matlab) to allow partner institutions to
  easily generate new curricula
• Write a methods paper explaining the models used
  in greater detail

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References

• Variable and Value Ordering When Solving Balanced
  Academic Curriculum Problems
• C. Castro and S. Manzano (2001)