1. Curriculum Optimization Project CURRICULUM REFORM MEETING California State University, Los Angeles Kevin Byrnes Dept. of Applied Math Johns Hopkins University

2. 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.

3. Model of the Problem as a Graph Model of the Problem as a Graph Topics correspond to vertices (nodes) Directed edges correspond to a direct prerequisite relationship between topics

6. 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.

8. 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.

9. 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)

10. Goals 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.

11. Variables for IP 1 x ij = 1 if topic I is assigned to course j, 0 otherwise. c i = the estimated number of credit hours needed to teach topic I t ij = the length of the shortest undirected path in G from topic I to seed j l b = lower bound on credit hours for a course u b = upper bound on credit hours for a course

13. 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 OF 1 attempts to assign each vertex to its closest seed (aka assign each topic to its ‘nearest’ course)

14. Interpreting the Output The output from IP 1 is a doubly- indexed vector x ij (aka. a matrix). We can interpret this output as a set of courses {S 1,…,S N } as follows: S j = the set of x ij for which x ij = 1

15. There & Back: Cycling and Other Dangers

16. Determining Prerequisity Simple Method: If topic S ag in course Sa is a prerequisite for topic S bh in course S b, the course S a is a prerequisite for course S b

17. 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.

18. An Alternate Method Test for Prerequisity: For two courses S a and S b, sum all the credit hours of distinct prerequisite topics in Sa, and distinct topics in S b having topics in S a as prerequisites. If both of these sums exceed some threshold values, then S a is a prerequisite for S b.

22. 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

24. The Feasibility Condition For any course S b, having course S a as a prerequisite, S b must be assigned to be taught during a semester no earlier than S a

25. Identifying Core Courses Having our set of courses, S 1,…,S N from IP 1, 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 IP 1 and look for ‘natural’ candidates. 2) Create an ‘essentiality’ index for courses.

26. Putting It All Together: Generating a Curriculum

27. 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

28. How 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

29. IP 2 variables x ij = 1 if course S i is assigned to semester j, 0 otherwise. a i = the expected number of credit hours it will take to teach course S i X ij denotes the decision variable for a required course l j and u j 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.

30. 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

32. Explanation of Constraints 1) defines d j 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 OF 2 minimizes the max over the number of credit hours assigned to each semester, and the penalty term

33. Output From IP 2 IP 2 gives us an optimal scheduling of courses for a specified value of alpha, epsilon, l j, and u j. 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 IP 2, and examine unsatisfactory prerequisite relationships in the optimal output

34. Where Do We Go From Here?

35. 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

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