1. Integrating Mathematical Thinking into the Curriculum: Examples
Dr. Christelle Scharff
Assistant Professor
Computer Science Department
New York
Presentation FRN NYU, June 2004

2. Outline Introduction Three examples
The Thinking through Computing Learning Community
Functional Programming in a Discrete Mathematics Course
WeBWorK for Discrete Mathematics
Workshop: Practice with WeBWorK

3. Integrating Mathematical Thinking into CS Curricula
Group created by D. Baldwin and P. Henderson in 1999
Mathematical Thinking: “Applying mathematical techniques, concepts and processes, either explicitly or implicitly, in the solution of problems -- in other words, mathematical modes of thought that help us to solve problems in any domain. In its most general interpretation, every problem-solving activity requires mathematical thinking. For example, basic logic, be it used explicitly or implicitly is required for all problem-solving activities.”

4. Highlightening the Connections between Maths and CS
Creation of a repository for instructors (and students) showing the connections between Maths and CS
Examples of connections:
Pre- and post-conditions, invariants, recursion, symbolic computations, relational algebra, formal specification and verification…

5. Pace University and Mathematical Thinking
CAFME (Center for the Advancement of Formal Methods in Education)
Use of Z in diverse undergraduate and graduate courses
Master in Software Design and Development
Software Validation and Verification
Mathematical Modeling of Software Artifacts

6. Hackers Know Logic!
“SQL Injection is a technique which enables an attacker to execute unauthorized SQL commands by taking advantage of unsanitized input opportunities in Web applications building dynamic SQL queries”
SELECT * FROM users WHERE username = ‘bob' and PASSWORD =‘bobpassword'
SELECT * FROM users WHERE username = ‘bob' and PASSWORD = ‘x’ or 1 = 1

7. The Thinking Through Computing Learning Community

8. Learning Communities
LCs: “Classes that are linked or clustered during an academic term, often around an interdisciplinary theme, and enroll a common cohort of students”
LCs as a way of solving problems linked with academic achievement and retention, and as models to import and export knowledge from one course to another
Interests of LCs for students:
Deeper understanding of materials by making connections
Learn to find similarities in disparate subject areas
Experience increased interaction with students and faculty
Participate in active and collaborative learning
Explore diverse perspectives
Interests of LCs for faculty

9. Courses and Connections
CIS 101: Introductory computing course for freshmen non-CS majors
Problem solving, algorithms, programming in Visual Basic, computer organization, data representation, networking, Excel, HTML, ethical issues in computing
PHI 253: Propositional Logic
Translate sentences from/to English to/from propositions, truth tables, formal proofs
Connections
Rigor in formal proofs and in programming
Transformation of English sentences to propositions, and then to Visual Basic statements
Truth tables in Excel
Advanced search in Google

10. Assignments
Programming in Visual Basic: The “Learn how to do truth tables” application that determines whether the user could construct truth tables for the OR, AND, IMPLIES and BICONDITIONAL connectives
Essay on social and ethical issues in computing
E-portfolio using HTML
Autobiographical presentation , LC description, reflective logs, links to students’ work…

11. Functional Programming and Discrete Mathematics

12. SUNY Stony Brook Curriculum
ACM Computing Curriculum
Sequence of courses:
Discrete Structure 1
Programming 1 Discrete Structure 2
Programming 2
Curriculum originally designed by Dr. Peter Henderson
Use of SML in Discrete Structure 1

13. Discrete Structure 1 Propositional logic Number theory Set theory
Functions
Recursion
SML (Standard Meta Language)
Induction
Automata
(Trees
Graphs)

14. Functional Programming Languages

15. Use of SML Students do not have programming experience
It is not a programming class. SML is introduced as a tool.
Why SML?
Non verbose syntax
Quick start
Subset of SML
Types and inference of types
Function definitions
Pattern matching
Higher order functions
Polymorphism

16. The reverse of a list

17. Interests Encourage mathematical thinking early in the curriculum
Convince students that the mathematics in discrete structures are relevant to their careers
Introduction to problem solving
Introduction to prototyping
Introduction to testing
Introduction to formal specification
Understanding of recursion

18. WeBWorK: An environment to Deliver Web-based Homeworks and More

19. What is WeBWorK?
More than an environment to deliver homeworks to students
Developed at the University of Rochester
Freely available to educational institutions
Mainly used in mathematics (algebra, calculus) and physics
Used by more than 50 institutions around the country including Columbia, Harvard, Stony Brook, Pace…
WeBWorK is extensible

20. WeBWorK for Students Easy of access (Internet)
Immediate feedback as to whether the answer is correct or not
Individualized version of the problems for each student
Each homework can be printed
Every problem has a Feedback button which sends an message directly to the instructor

21. The WeBWorK Professor Management of the course
List of students, passwords
Add and remove students
Monitor the progress of the students
Create and manage an exam
Create and deliver homeworks
Starting date, due date
Automated grading
Automated delivery of the solution

22. Problem Sets Create problem sets with solutions
Written in PG (Problem Generating)
Perl, html, latex
Individualized version of the problems for each student
Questions: matching, multiple-choice…
“Ask the question you should rather than the questions you can”

23. Use of WeBWorK
More than 1000 students used WeBWorK at SUNY Stony Brook in a Discrete Mathematics course
Problems on number theory, set theory, function, recursion and SML
WeBWorK is interfaced with Oliver to deliver propositional logic problems
50 students used WeBWorK at Pace University for Functional Programming and Logic Programming problems

25. Workshop: Practicing with WeBWorK

26. URL: http://webwork.csis.pace.edu/webwork/cs361/
Login: 
Username: practice1 to practice26
Practice1 to 4: The due date is passed.
Practice 5 to 26: The due date is: 6/21/04.
Password:
Choose your own password

27. Examples of problem sets:
Please, discard the error messages that may appear in the bottom of the pages!
Examples of problem sets:
Set 1 – Exercises on Recursion
Set 2 – Exercises on SML
Set 3 – Exercises on Logic Programming
Set 22 – Exercises on Number Theory
Set 44 – Exercises on Functions and Recursion

28. Interesting Problems:
Set 1 - Problem 1 – RecursionEval1.pg
Set 2 - Problem 2 – smltypes2.pg
Set 3 - Problem 1 – logic.pg
Set 22 - Problem 3 – decimalToRational.pg
Set 44 - Problem 4 - composition1.pg

29. References http://learningcommons.evergreen.edu