Insights from Teaching Discrete Mathematics in IS Programs – Discussion Forum Report
Harvey/Holdan, RMU, TFM Symposium, November 2004, Ghent, Belgium – Page 1
Insights from Teaching Discrete Mathematics
in Information Systems Programs
A Report for the Discussion Forum, 19 November 2004
CoLogNet/Formal Methods Europe Symposium on Teaching Formal Methods
Valerie J. Harvey, RT(R), PhD, Computer and Information Systems
E. Gregory Holdan, PhD, Mathematics Education
Robert Morris University, Moon Township, PA 15108 USA
Abstract
This report offers recommendations on how to assure a more educational effective role for the
basic skills and understandings required for the computing disciplines, including formal
methods. The recommendations, based on experiences developing a discrete mathematics
curriculum for information systems programs, address the teaching of discrete mathematics,
articulation with other disciplines at a variety of levels to gain more attention for formal
structures, and the specification, development, and marshalling of materials and software
which will support discrete mathematics teaching.
Keywords: discrete mathematics, information systems, formal methods, mathematics
education
1.
INTRODUCTION
This report covers the effort to determine
what discrete mathematics topics are
relevant for information systems students,
what teaching methods are suitable, and
what teaching resources (materials and
software) are necessary for effective
instruction.
According
to
Kilov,
mathematics “provides not only the
foundation needed to ‘feel safe’ in
specifying semantics but also patterns of
reasoning essential for understanding of
any complex phenomena” and since
“computer-based information systems are
dealing with discrete phenomena and
discrete structures (for such systems it is
generally incorrect that a small change in
inputs will lead to a small change in
outputs), the mathematics used to model
and understand such systems is discrete
mathematics.” [KIL2003]
The announcement for the CoLogNet/FME
Symposium on Teaching Formal Methods
focuses on how the teaching of formality
and mathematics needs to be better
received and supported in education.
[FME2004]
Discrete
mathematics
is
routinely supported in certain programs
beyond
mathematics
(as
discrete
mathematics or symbolic logic) and
philosophy (as introductory logic or
symbolic logic). The teaching of discrete
mathematics, which provides the basis for
formal methods, is traditional in computer
science (CS) and software engineering
(SE) and is specified by the Accreditation
Board for Engineering and Technology
(ABET) computer science accreditation
criteria in the U.S. In the September 2003
issue of Communications of the ACM
dealing with mathematics, Keith Devlin,
Kim Bruce, Robert Drysdale, Charles
Keleman,
Allen
Tucker,
and
Peter
Henderson all find that the form of
mathematics
of
particular
value
to
computer scientists and software engineers
is
discrete
mathematics.
[BRU2004,
DEV2003, HEN2003] The report of the
ITICSE 2000 Working Group on Formal
Methods Education cites the role of discrete
mathematics in formal methods and
asserts that “without formal methods, the
[software] engineering is non-existent or
suspect.” [ALM2000]
A number of information systems (IS)
curricula require enrollment in discrete
mathematics or include it as an option
within the set of requirements. With the
recent development of ABET accreditation
for
information
systems,
information
system programs in the U.S. now have
Insights from Teaching Discrete Mathematics in IS Programs – Discussion Forum Report
Harvey/Holdan, RMU, TFM Symposium, November 2004, Ghent, Belgium – Page 2
substantial encouragement to include
discrete mathematics in the curriculum.
[ABE2003]
2. RELEVANCE OF FORMAL
STRUCTURES
Discrete
mathematics
topics
support
judgment and decision-making in a variety
of fields. A few of the practical impacts of
discrete mathematics for persons working
with information technology and systems
include: understanding the generality of
graphs so they can better interpret and
more effectively use graphs in technical
areas, such as networking and databases;
designing correct implementations of rules
for applications; designing databases and
correct database queries; generating test
data; understanding various codes (such
as Huffman codes and Gray codes) and
their applications; and understanding state
transition diagrams in general and applying
them with regard to operating systems and
project management. Students should
come to value the significance of formal
structures for information technology.
Wildstrom pointed out the relevance of
graph
theory
for
Web
technology.
[WIL2002]
The computing disciplines require a
different
configuration
of
discrete
mathematics topics than is common for
discrete
mathematics
courses
which
primarily
serve
students
studying
mathematics. In developing a discrete
mathematics course for IS, a matrix
(discrete mathematics topics vs. core
curriculum topics) has been used to
determine
relevance
of
discrete
mathematics topics for a local ABETaccreditable IS curriculum and as a basis
for identifying practical applications of
discrete math. [HAR2004] In meeting the
needs of CS, the ACM/IEEE Computing
Curriculum 2001: Computer Science Body
of Knowledge has been used to determine
which discrete mathematics topics are
needed for CS. [ACM2001, DEC2004]
As a discrete mathematics course was
being designed for IS students, there were
questions about whether formal logic
notation should be used in a course
intended for information systems and
information systems management majors.
Obviously internationally accepted formal
notations make the material clearer and
easier
to
learn.
Formal
structures
represent only part of the task of teaching
discrete mathematics. Much of the effort
involves
understanding
how
logical
structures are communicated in everyday
natural language. Ambiguities in natural
language,
as
in
specifications
for
applications, systems, and projects, must
be addressed. Formal notation is of great
practical help in this process of learning
about the meanings of specifications.
Students readily apply what they have
learned using formal notation to such
practical concerns as database design,
conduct and management of programming
in
application
development,
and
understanding the properties of data
communications networks. To help explain
the importance of formal notation, we
pointed out how unnecessarily challenging
it would be to teach music while avoiding
use of internationally accepted music
notation. Information system students
report that their discrete mathematics
experience, presented to them using
formal notation, helps them comprehend
specifications (as they would need to do in
the system analysis process in their
careers).
They
experience
increased
correctness and confidence in such tasks.
The IS discrete mathematics course omits
or reduces the coverage of some discrete
mathematics topics through the matrix
process. In a few cases the result regarded
as appropriate for information systems
goes beyond the traditional approach. The
definition of partial order includes coverage
of strict partial order rather than just the
usual (reflexive) characterization because
practical IS examples generally involve
strict partial order (such as the order of
creating tables and loading data (and
inversely of dropping tables) in relational
database designs with enforced referential
integrity). We also found that a treatment
of relations and functions accounting
explicitly for source and target sets and
clearly distinguishing them from domain
and range, as in formal methods (FMs),
meets IS teaching needs. The distinction
between source set and domain (or target
set and range) has practical implications
for the design of relational database (SQL)
queries in database management courses.
Interdisciplinary development was helpful
in
evaluating
curriculum
design
Insights from Teaching Discrete Mathematics in IS Programs – Discussion Forum Report
Harvey/Holdan, RMU, TFM Symposium, November 2004, Ghent, Belgium – Page 3
alternatives. Decker and Ventura have
described the particular needs of CS
students studying discrete mathematics.
[DEC2004]
Note that since the career path of IS
graduates may bring them to management
positions in IT organizations, IS graduates
may make the decisions about whether
FMs are utilized in a given project. If IS
majors increasingly receive a foundation in
discrete math, they may be more likely to
value the benefit of FMs in various
industries. We estimate there are about
1,000
undergraduate
programs,
approximately 2,000 associate degree
programs and about 350 masters’ degree
programs in the U.S.
We can work to include the strategies,
tools, and notation of FMs in discrete
mathematics curricula and textbooks so
that
discrete
mathematics
courses,
especially those intending to support
computing disciplines, increasingly become
consistent with FMs in concepts and
notation. Some discrete mathematics texts
already have chapters or units in FMs.
Edmond provides an example of a text that
integrates the treatment of databases,
SQL, and ER modeling with presentation of
formal
information
modeling
and
specification. [EDM1992]
If a productive interdisciplinary working
relationship
exists
between
IS
(or
whatever department offers the IS
program, or CS) and the mathematics
department, then a discrete mathematics
course that is truly appropriate for IS or CS
students can be achieved. With the right
relationship, mathematics faculty can work
with IS or CS faculty to fashion modules
for the computing discipline courses to
illustrate discrete math concepts in
application context and IS or CS faculty
can support math faculty with IS/CSrelevant and technology-relevant examples
and information for discrete math teaching
of students in the particular computing
disciplines.
Adequate
exchange
of
information then may mean that the
departmental location of the discrete
mathematics course is not important, or
that the course can be cross-listed and
taught by faculty from either department.
3.
FORMAL STRUCTURES AT
DIFFERENT LEVELS OF EDUCATION
Within the scope of communications and
natural language instruction (in English,
French, German, Flemish, Dutch, Spanish,
Chinese,
and
others),
modules
in
communication skills courses can identify
and emphasize the logical and semantic
properties of certain queries so that
individuals learn to interpret meaning
correctly or to recognize when more
information is needed to determine
meaning. The meaning of queries can be
explored without any use of computers.
Where such modules are in place, it may
productive to show that the skills being
acquired are of use not only in general
discourse, but even in technical work
environments,
such
as
database
programming. [HAR2003a]
High school mathematics programs and
college core or general studies curricula
can include an introduction to logic
(possibly
within
the
context
of
a
communication skills program) so that
students become acquainted with the
concepts
of
set
difference
and
complements and with set operations in
general. In some countries all students
receive logic instruction covering sets, set
operations,
and
predicate
logic.
Exclusionary relational database queries,
for
example,
require
not
only
understanding of SQL, but also correct
comprehension of negation in the natural
language query and correct interpretation
of
the
logic
involved.
[HAR2003a]
Recommendations have been made for
discrete
mathematics
courses
and
instruction
at
the
secondary
level.
[HAR1990, ROS1997]
4. TEACHING STRATEGIES AND
TECHNIQUES
The Colognet/FME program announcement
acknowledges the challenges of “phobia.”
The teaching of discrete mathematics can
be more effective and better received with
attention to motivation and practical
applications. Such approaches are not new.
[KAR1940, EVY1981] There are many
treatments of how to motivate learners in
mathematics
through
attention
to
applications and such recommendations
are embedded in the “General Strategies
Insights from Teaching Discrete Mathematics in IS Programs – Discussion Forum Report
Harvey/Holdan, RMU, TFM Symposium, November 2004, Ghent, Belgium – Page 4
for Software Engineering Pedagogy,” of the
ACM/IEEE Software Engineering 2004
Curriculum Guidelines for Undergraduate
Degree Programs in Software Engineering,
especially in curriculum guideline 16.
[ACM2004]
Instructional strategies useful in teaching
discrete
mathematics
to
information
systems students include: (1) determining
the relevance of discrete mathematics
topics
through
matrix
of
discrete
mathematics topics  core curriculum
courses and topics; (2) introducing discrete
mathematics as relevant by showing
examples of use of formal structures in
information technology and applications
(i.e. various kinds of graphs and trees from
networking; transition diagrams from
operating systems, Venn diagrams from
wireless protocols, set operations from
video production software, Gantt charts
from project management, logic in
processor circuits, automata as the basic
building blocks of software, etc.); (3)
selecting exercises that are applicationoriented and that are interesting to them
because of an information technology
connection or application where possible;
(4) building on, and reinforcing, students’
backgrounds (beginning in grade school) in
discrete mathematics and experience and
affirm what students already know about
discrete mathematics when they enter the
course – seek confidence, motivation, and
pride in learning; (5) showing how the
knowledge and skills students are acquiring
in the course help them in current studies
and can help them in their career roles and
career
role
decision-making
about
technology (view the students as the
information technology decision-makers of
the future); and (6) providing students
with excellent interactive software learning
experiences – giving them tools for
exploration and self-directed learning.
Current findings from the 1995 TIMSS
(Trends in International Mathematics and
Science Study) video studies suggest very
strongly that mathematics teachers need
to implement instructional strategies that
go
above
and
beyond
teaching
procedurally and algorithmically, with a
primary focus on getting correct answers.
Instead, teachers need to provide students
with instructional experiences that facilitate
learning through making conjectures and
testing through exploration. [STI1999,
TIM1995] Some assignments used as
springboards for instruction need to be
“doing” mathematics type problems, where
students
experience
some
cognitive
dissonance from not having at-hand readyto-use procedures that yields a correct
answer. Mathematical tasks with a focus on
a given concept should have interesting
connections to other mathematics concepts
and to real life. Tasks should be inherently
interesting and worthy of “mining” for the
implicit mathematics. Teachers need to
capitalize on multiple representations of
problems and their solutions, provide
opportunities for students to share their
thinking, and capitalize on teachable
moments that naturally arise during a real
problem-solving experience. Examples of
alternative representations in the domain
of discrete mathematics obviously include
comprehending and manipulating graphs
as adjacency and incidence matrices,
recognizing equivalent expressions in
different notation patterns (infix and
prefix) and equivalent logic statements
(using the principles of commutivity and
association, restatement of conditionals as
disjunctions, and DeMorgan’s Laws, and
expressed
in
natural
language,
propositional or predicate logic, Boolean
algebra, or circuit diagrams).
It is vital that students be able to restate
formal expressions (out loud or in writing)
as statements in natural language or vice
versa. In discrete mathematics this means
designing exercises and various kinds of
assessments to be certain, at each stage of
learning, that students can render, for
example, “x (T(x)  P(x))” (which might
be understood as “all trespassers will be
prosecuted”) or expressions using set
builder notation, in natural language. Once
students can say or write the formal
expressions in natural language, they can
study more effectively and confidently on
their own and in small group sessions with
other students. Such capability needs
explicit attention in the curriculum design.
Humor helps (“For every upside down ‘A’
there is at least one backwards ‘E,’” or
“there are 10 kinds of people, those who
understand binary and those who don’t”).
Practice with natural languages leads
conveniently to practical exploration and
exercises, such as evaluating the logic of
health care insurance specifications for
Insights from Teaching Discrete Mathematics in IS Programs – Discussion Forum Report
Harvey/Holdan, RMU, TFM Symposium, November 2004, Ghent, Belgium – Page 5
reimbursement
or
of
labor
law
specifications for overtime and proposing
use in any given programming language to
design, implement, and test a rule.
The move from treatment of individuals to
generalization (and various categories of
variables) is an important step in certain
discrete mathematics topics (particularly
statements with quantification in predicate
logic). Special care is needed to assure
that every learner in a group acquires the
capability to understand and use variables.
Instructors of discrete mathematics can
benefit from engaging in group “faculty
development” activities (in departments or
at conferences) such as collaborative
lesson plan creation, maintenance and
revision of exemplary lesson plans and
presentations, observing other instructors
teaching lessons, sharing of experiences
with analogies, practical examples, and use
of teaching tools, and engaging in
retrospective discussion of a given lesson
presentation as it unfolded in the
classroom. [HOL2004a, HOL2004b]
5. TEACHING RESOURCES
Increasingly
discrete
mathematics
textbooks used for information systems
programs should replace some of the
traditional examples and exercises in
discrete mathematics topics with practical
examples and exercises drawn from the
various
disciplines
associated
with
computer technology: operating systems,
data communications and networking,
network and computer security, database
management, and programming logic for
application development.
Since discrete mathematics is important in
a number of curricula and it vital to
support disciplines relating to information
technology, the learning of discrete
mathematics (set theory, graph theory,
logic
and
predicate
calculus,
state
transition diagrams) should be adequately
served by instructional software at a
variety of levels.
It should be convenient for a student or an
instructor to generate and use a Venn
diagram, overbar negation or complement
notation, to generate and edit and to
demonstrate traversal of a directed graph
(with adjacency matrix articulation) or of a
tree, to generate and edit and to
demonstrate a state transition diagram to
illustrate process scheduling in a particular
operating system, to generate and edit a Z
or VDM model-based specification, Hasse
diagrams, relational algebra or calculus,
BNF, Petri nets, or many other relevant
forms of notation. Since graph coloring can
be used to represent and solve many
different
scheduling
problems,
graph
coloring capability should be included.
[LIL2003, SCO2004]
Microsoft Equation® and Decision Science
MathType® should be adequate and
convenient
for
common
discrete
mathematics needs. It should be possible
to create Web pages easily with Venn
diagrams, graphs, and various kinds of
discrete mathematics notation and publish
them on the Web. Students should be able
to easily manipulate symbols and graphs
and include them in their e-mail messages.
Current
products
should
support
applications
of
discrete
mathematics
(where labeling requirements may diverge
from traditional labeling.
We have excellent teaching software to
support some discrete mathematics topics
at a variety of levels. [BAR2002, HAC2002,
PEM2003, ROD2004, COL2004] If we
inquire and explore systematically, we will
find that at least certain important areas of
discrete mathematics do not receive the
same level of focus and priority as other
forms of mathematics in the availability of
interactive
instructional
software
for
editing, manipulation, practice, simulation,
and exploration. A number of discrete
mathematics activities are supported by
major products like Mathematica™ and
Maple™, but these may use notations
different from those found in the textbooks
or may in other ways be challenging for
students beginning the study of discrete
math, although valuable for intermediate,
advanced and exploratory levels. Discrete
mathematics resources may be scattered
in independent applications, rather than
provided in an integrated environment. If
discrete mathematics is as important as is
indicated in the CACM articles, then
perhaps we should find a way to encourage
priority support for discrete mathematics in
the major products and in the Microsoft™
environment and we should also act to
Insights from Teaching Discrete Mathematics in IS Programs – Discussion Forum Report
Harvey/Holdan, RMU, TFM Symposium, November 2004, Ghent, Belgium – Page 6
assure that any needed resources lacking
are made available.
We can work together with the other
disciplines that have an interest in discrete
mathematics. Besides formal methods
(FMs),
this
means
philosophy,
mathematics,
computer
science,
information systems, software engineering,
and information technology (IT) on
curriculum and resources. A large number
of disciplines will have more opportunity to
obtain adequate support for development
of
software
resources.
Acting
in
collaboration with the disciplines that teach
and incorporate discrete math, we could
assess the possibility of developing a
project to make sure significant discrete
mathematics
needs
for
instructional
software are met. It is important that any
project not duplicate existing efforts and
also articulate with the many existing
resources, projects, and repositories.
Here is the process proposed:
Systematically, comprehensively, and on
an interdisciplinary basis, we should assess
the needs for instructional software for
discrete mathematics, shared by several
disciplines, and including/involving major
textbook authors if possible. to determine
what is required to support teaching in
discrete mathematics for the study of:
Computer Science, Software Engineering,
Information Systems, Information Systems
Management and Management Information
Systems, Formal Methods, Information
System Technology, mathematics, and
philosophy. Software is needed for four
levels
of
curriculum/instruction:
introductory,
intermediate,
advanced
exploration, and research applications.
Assessment should include criteria and
requirements specification for interactive,
manipulation-oriented, simulations; editing
capabilities; ease of use; saving of files;
printing of states and results; ease of
publishing results online (Web); ease of
use in e-mails; drawing or generating
graphics; labeling true to standard
notations (note overbar notation for
complements, negation); appropriateness
and adequacy of labeling, notation, format
– suitable for application needs (e.g. Hasse
diagrams for relational database designs,
finite state diagrams for operating systems
(process states), etc.)
Discrete mathematics teaching software
should support alternative representations
and notations, for example graphs as
adjacency and incidence matrices, graphs
as sets of ordered pairs, and circuits as
Boolean algebra expressions. It should be
convenient to cycle between alternative
representations.
Some products
now
provide this.
Attention must be given to availability for
instruction with regard to licensing, cost,
environments (PC, Mac, Linux/Unix), and
support for presentations in different
natural languages.
The process of assessment should include
interdisciplinary
representation
and
representation of major authors of discrete
mathematics textbooks.
The first step should be identification and
review/evaluation of existing software for
adequacy and classification (and crossindexing) of products as to discrete
mathematics
topic(s)
supported
and
adequacy for different levels of instruction.
We should set a priority on discrete
mathematics presentations which students
recognize are the most important and
valuable for them.
This process should acknowledge and use
all existing organizational and individual
catalogs of discrete mathematics resources
and should cover (1) current teaching
products such as Wolfram Research
Mathematica and Combinatorica, Maplesoft
Maple, and MathWorks MATLAB, CSLI
Tarski’s World, etc.); (2) documentation
production,
graphics
editing,
and
mathematics notation tools such as
Microsoft (notation, diagrams), Microsoft
Visio, Decision Science MathType, and
SmartDraw.
Then there should be a determination of
software (or software features) needed and
not available. The third step would be
determination of how to meet needs not
currently met.
Next, software can be obtained by
initiatives of educators, researchers, or
current vendors, or by commissioning it
through jointly-sought grant or project
support. Articulation of needs to existing
vendors could lead to revision or upgrade
of current products or to new independent
products in a given environments.
Insights from Teaching Discrete Mathematics in IS Programs – Discussion Forum Report
Harvey/Holdan, RMU, TFM Symposium, November 2004, Ghent, Belgium – Page 7
Finally there should be development and
implementation
of
strategies
for
dissemination and training, using existing
centers and institutions. Cooperating sites
can provide online listings, including Web
sites and downloads for software, classified
by discrete mathematics topic.
6. OUTCOMES ASSESSMENT
Logic
is
a
key
part
of
discrete
mathematics. With regard to discrete
mathematics, the “Propositional Logic Test
(PLT)
can
be
used
for
outcomes
assessment. The PLT, developed and used
over a number of years by science
education faculty and students at Rutgers
University, assesses an individual's ability
to process propositional statements. The
PLT is a timed task in which the individual
interprets truth-functional operations by
identifying instances that are consistent or
inconsistent with a stated rule… .”
[ALM1996]. The PLT needs automation,
including a validating of the proposed
resulting system, so that educationally
acceptable rearranged versions can be
generated to support more flexible use and
scoring and interpretation can be made
easier.
Since outcomes assessment is important to
broader
incorporation
of
discrete
mathematics in curricula at a variety of
levels, additional discrete mathematics
assessment instruments need to be
specified,
designed,
implemented,
validated, and disseminated.
Assessment within a course should go
beyond seeking evidence of student
learning to provide feedback that supports
both immediate and long-range refinement
and improvement of teaching. [BLA2004]
7. CONCLUSION
Appropriate
discrete
mathematics
instruction enhances education in the
computing disciplines by providing a sound
formal foundation for and insights into an
increasingly
complex
information
technology, helping students recognize and
use practical problem-solving tools, and
helping professionals develop insights
useful in making management decisions
with regard to applications of formal
structures in business and industry.
Discrete
mathematics
supports
the
capability of professionals to contribute to
software reliability, safety, and suitability
in
information
technology.
Discrete
mathematics instruction can be made more
effective through attention to applications
and
relevance,
textbooks
that
are
application-oriented,
and
engaging
interactive
software
experiences.
Collaboration among the disciplines with an
interest in discrete mathematics can lead
to joint initiatives to gain more attention to
discrete
mathematics
topics
across
secondary and undergraduate education
and to strong interdisciplinary support for
identification and development of teaching
software resources that are in the common
interest.
8. ACKNOWLEDGEMENTS
Guidance, insights, and input from the
following
persons
are
gratefully
acknowledged:
Vicki
L.
Almstrum,
University of Texas; Peter B. Henderson, ,
Butler University; Charles Hacker, Griffith
University, Queensland, Australia; Herbert
E. Longenecker, Jr., University of South
Alabama; Dave Barker-Plummer, Center
for
the
Study
of
Language
and
Information, Stanford University; Steve
Wildstrom, Technology & You Editor,
BusinessWeek; Susanna S. Epp, DePaul
University; Frank E. Ritter, Applied
Cognitive
Science
Lab,
Penn
State
University; Judith L. Gersting, University of
Hawaii at Hilo, Hilo, HI; Joe Mott, Florida
State University; Ada C. Dong, Lawrence
Technological
University;
Mary-Angela
Papalaskari, Villanova University; Kenneth
A. Lloyd, Jr., Watt Systems Technologies,
Inc.; Susan Rodger, Duke University; Doug
Baldwin, SUNY Geneseo; Haim Kilov,
Independent Consultant; Elsje Scott,
University of Cape Town; Anthony M.
Lopez, Jr., Xavier University of Louisiana;
Roy Daigle, University of South Alabama;
Richard Botting, California State University.
9. REFERENCES
[ABE2003]
Criteria
for
Accrediting
Information Systems Programs (Effective
for Evaluations during the 2004-2005
Accreditation
Cycle),
Computing
Accreditation Commission, Accreditation
Board for Engineering and Technology,
Inc., approved November 1, 2003.
Insights from Teaching Discrete Mathematics in IS Programs – Discussion Forum Report
Harvey/Holdan, RMU, TFM Symposium, November 2004, Ghent, Belgium – Page 8
[ACM2001] ACM/IEEE Joint Task Force on
Computing
Curricula,
Computing
Curriculum 2001: Computer Science.
http://www.computer.org/education/cc200
1/cc2001.pdf, see Appendix A: CS Body of
Knowledge.
[ACM2004] ACM/IEEE Software
Engineering 2004 - Curriculum Guidelines
for Undergraduate Degree Programs in
Software Engineering:
http://www.computer.org/education/cc200
1/SE2004Volume.pdf
[ALM<no date>] Almstrum, Vicki L.,
Limitations in the Understanding of
Mathematical Logic by Novice Computer
Science Students. Dissertation, University
of Texas.
[ALM1994] Almstrum, Vicki L., “Some
Background on the Propositional Logic
Test,” Department of Computer Sciences,
The University of Texas at Austin. URL:
almstrum@cs.utexas.edu
[ALM1996]
Almstrum,
Vicki
L.,
Investigating student difficulties with
mathematical logic. In N. Dean and M.
Hinchey (eds.), Teaching and Learning
Formal Methods, Academic Press, 1996.
[ALM1999] Almstrum, Vicki L., The
Propositional Logic Test as a Diagnostic
Tool for Misconceptions about Logical
Operations. Journal of Computers in
Mathematics and Science Teaching. 18(3).
1999.
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