WeBWorK - Pace University

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Integrating Mathematical
Thinking into the
Curriculum:
Examples
Dr. Christelle Scharff
cscharff@pace.edu
Assistant Professor
Computer Science Department
New York
Presentation FRN NYU, June 2004
1
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
2
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 problemsolving activities.”
• http://cs.geneseo.edu/~baldwin/math-thinking
3
Highlighting 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…
• http://blue.butler.edu/~phenders/iticse2002
4
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
• Try it at:
http://www.csis.pace.edu/~scharff/sqlinjection
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
The Thinking Through
Computing Learning
Community
Pace University
Fall 2002
7
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 achievements
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
•
http://learningcommons.evergreen.edu
8
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
9
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
• Writing an essay on social and ethical issues in
computing
• Designing an E-portfolio using HTML
• Autobiographical presentation , LC description,
reflective logs, links to students’ work…
10
Functional Programming
and Discrete Mathematics
SUNY Stony Brook
Fall 2000 and Spring 2001
11
SUNY Stony Brook
Curriculum
• ACM Computing Curriculum 2001
http://www.acm.org/sigcse/cc2001
• Sequence of courses:
Discrete Structure 1
Programming 1
Prerequisite
Discrete Structure 2
Programming 2
• Curriculum originally designed by Dr. Peter
Henderson
• Use of SML in Discrete Structure 1
12
Discrete Structure 1
•
•
•
•
•
•
•
•
•
•
Propositional logic
Number theory
Set theory
Functions
Recursion
SML (Standard Meta Language)
– http://www.smlnj.org/ and SML .NET
Induction
Automata
(Trees
Graphs)
13
Functional Programming
Languages
14
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
15
– Polymorphism
Factorial
fun fact(n) =
Without Pattern
Matching
With Pattern
Matching
if n = 0 then 1
else n * fact(n-1);
fun fact(0) = 1
| fact(n) = n * fact(n-1);
16
The reverse of a list
Without Pattern
Matching
With Pattern
Matching
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
SUNY Stony Brook
Spring 2001
Pace University
Fall 2001 and after
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
• http://webwork.math.rochester.edu
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 e-mail message directly to the
instructor
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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
• http://www.csis.pace.edu/~scharff/SOFTWARE/OLIVER/o
liver.html
• 70 students used WeBWorK at Pace
University for Functional Programming and
Logic Programming problems, and in the
Thinking Through Computing LC
24
An example of problem
and the PG code
25
Question 1 – Source
code
# A question requiring a string answer.
$string = 'world';
BEGIN_TEXT
Complete the sentence: $BR
\{ ans_rule(20) \} $string;
$PAR
END_TEXT
ANS( str_cmp( "Hello") );
26
Question 2 – Source
code
# A question requiring a numerical answer.
#define the variables
$a=random(1,9,1);
$b=random(2,9,1);
BEGIN_TEXT
Enter the sum of these two numbers: $BR
\($a + $b = \) \{ans_rule(10) \} $PAR
END_TEXT
$sum = $a + $b;
ANS( num_cmp( $sum ) );
27
Question 3 – Source
code
# A question requiring an expression as an
answer
BEGIN_TEXT
Enter the derivative of \[ f(x) = x^{$b} \] $BR
\(f '(x) = \) \{ ans_rule(30) \}
$PAR
END_TEXT
$new_exponent = $b-1;
$ans2 = "$b*x^($new_exponent)";
ANS( fun_cmp( $ans2 ) );
28
Workshop: Practicing
with WeBWorK
29
• 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
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• 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
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• 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
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References
•
•
•
•
•
http://learningcommons.evergreen.edu
http://www.cs.geneseo.edu/~baldwin/math-thinking
http://webwork.math.rochester.edu/
http://www.acm.org/sigcse/cc2001
A. Wildenberg and C. Scharff. Oliver: an OnLine Inference and
VERification system. FIE 2002.
• C. Scharff and A. Wildenberg. Teaching Discrete Structures with
SML. FDPE 2002.
• C. Scharff and A. Wildenberg. Supporting Discrete Structures
Courses with a web-based tool. SIGCSE 2003.
• C. Scharff and H. Brown. Thinking Through Computing: The
Power of Learning Communities. 2004.
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