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Curriculum Optimization Project


2) The curriculum should allow flexibility. 3) The new curriculum should be completable in the same time needed for the current one. ... – PowerPoint PPT presentation

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Title: Curriculum Optimization Project

Curriculum Optimization Project
  • California State University, Los Angeles
  • Kevin Byrnes
  • Dept. of Applied Math
  • Johns Hopkins University

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

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
  • We can estimate the number of seeds we want by
    the number of courses currently in the curriculum.

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)

Goals in Constructing a Course
  • We would like to assign each topic to its nearest
    seed (course) for intellectual cohesiveness (goal
  • 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.

Variables for IP1
  • xij 1 if topic I is assigned to course j, 0
  • 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

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

There BackCycling and Other Dangers
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

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.

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

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.

Putting It All TogetherGenerating a Curriculum
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

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

IP2 variables
  • xij 1 if course Si is assigned to semester j, 0
  • ai the expected number of credit hours it will
    take to teach course Si
  • Xij denotes the decision variable for a required
  • 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.

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
  • 4) bounds the number of credit hours assigned to
    a semester above and below
  • 5) bounds the number of courses assigned to a
  • OF2 minimizes the max over the number of credit
    hours assigned to each semester, and the penalty

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

Where Do We Go From Here?
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

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