Curriculum Optimization ProjectCurriculum Optimization Project

CURRICULUM REFORM MEETING

California State University, Los Angeles

Kevin Byrnes

Dept. of Applied Math

Johns Hopkins University

Project GoalsProject 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.

Model of the Problem as a Model of the Problem as a GraphGraphTopics correspond to vertices

(nodes)Directed edges correspond to a

direct prerequisite relationship between topics

A Shortest PathA 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.

From Topics to CoursesFrom 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.

How to identify seedsHow 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 CourseGoals in Constructing a Course

We would like to assign each topic to its nearest seed (course) for intellectual cohesiveness (goal #1)

We’ll 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 IPVariables for IP11

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

Explanation of ConstraintsExplanation 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)

Interpreting the OutputInterpreting 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 & Back:There & Back:Cycling and Other DangersCycling and Other Dangers

Determining PrerequisityDetermining 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 DifficultiesSome 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 MethodAn 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.

Another Pitfall: Directed CyclesAnother 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

The Feasibility ConditionThe 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 CoursesIdentifying 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 Together:Putting It All Together:Generating a CurriculumGenerating a Curriculum

Goals for the CurriculumGoals 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 CurriculumHow to Construct the Curriculum

We’ll 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

IPIP22 variables 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.

The PenaltyThe 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

Explanation of ConstraintsExplanation 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

Output From IPOutput From IP22

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?Where Do We Go From Here?

The Next StageThe 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

ReferencesReferences

Variable and Value Ordering When Solving Balanced Academic Curriculum Problems

C. Castro and S. Manzano (2001)