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
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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
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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
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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
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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);
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The reverse of a list
Without Pattern
Matching
With Pattern
Matching
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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
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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
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• 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
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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”
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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
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An example of problem
and the PG code
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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 ) );
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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 ) );
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Workshop: Practicing
with WeBWorK
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• 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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