Isecondmwkshp1 - Robert Morris University



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 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 ABET-accreditable 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 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/CS-relevant 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 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 application-oriented 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 ready-to-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 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 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 cross-indexing) 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.

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.

[ACM2001] ACM/IEEE Joint Task Force on Computing Curricula, Computing Curriculum 2001: Computer Science. , see Appendix A: CS Body of Knowledge.

[ACM2004] ACM/IEEE Software Engineering 2004 - Curriculum Guidelines for Undergraduate Degree Programs in Software Engineering:

[ALM] 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.

[ALM2000] Almstrum, Vicki, Dean, C. Neville, Goelman, Don, Hilburn, Thomas B., and Smith, Jan, “Support for Teaching Formal Methods, Report of the ITiCSE 2000 Working Group on Formal Methods Education.”

[BAR2002] Barwise, Jon, and Echtemendy, John (in collaboration with Allwein, Gerard, Barker-Plummer, Dave, and Liu, Albert), Language, Proof, and Logic (CSLI, 2002) This book is published as a package with the software for Tarski's World, Fitch, and Boole.

[BLA2004] Black, P., Harrison, C., Marshall, B., and Wiliam, D., “Working Inside the Black Box: Assessment for Learning Inside the Classroom,” Educational Leadership 86, 1 (2004): 9-21.

[BRU2003] Bruce, Kim B., Drysdale, Robert L. Scot, Keleman, Charles, and Tucker, Allen “Why Math?” CACM 49, 9 (September 2003): 40-44.

[COL2004] Collberg, Christian, Kobourov, Stephen G., and Westbrook, Suzanne, “Algovista: an Algorithmic Search Tool in an Educational Setting,” Proceedings of 35th ACM SIGCSE Symposium, Norfolk, VA, 2004, pp. 462-466.

[CUP2004] CUPM Curriculum Guide 2004: Undergraduate Programs and Courses in the Mathematical Sciences. Mathematical Association of America, 2004.

[DEA1996] Dean, C. Neville, and Hinchey, Michael G., eds., Teaching and Learning Formal Methods (Academic Press, 1996).

[DEC2004] Decker, Adrienne, and Ventura, Phil, “We Claim this Class for Computer Science. A Non-Mathematician’s Discrete Structures Course,” Proceedings of 35th ACM SIGCSE Symposium, Norfolk, VA, 2004, pp. 442-446.

[DEV2003] Devlin, Keith, “Why Universities Require Computer Science Students to Take Math,” CACM 49, 9 (September 2003): 37-39.

[EDM1992] Edmond, David, Information Modeling: Specification and Implementation (Prentice Hall, 1992).

[EVY1981] Evyatar, A., and Rosenbloom, P., Motivated Mathematics (Cambridge University Press, 1981), p. vi.

[FME2004] “Symposium Announcement and Preliminary Program,” CoLogNet / Formal Methods Europe Symposium on Teaching Formal Methods 2004. See

[GRA1996] Grassmann, Winfried Karl, and Tremblay, Jean-Paul, Logic and Discrete Mathematics: A Computer Science Perspective (Prentice Hall, 1996), Chapter 8, “The Formal Specification of Requirements in Z.”

[HAC2002] Hacker, Charles, and Sitte, Renate, “A Computer-based Interactive Teaching Software for the Tracing of Logic Levels in a Digital Circuit,” Global Journal of Engineering Education 6, 1 (2002), 85-90.

[HAR1990] Hart, Eric W. et al., "Teaching Discrete Mathematics in Grades 7-12," Mathematics Teacher 83, no. 5 (1990): 362-367. See .

[HAR2003a] Harvey, Valerie J., Baugh, Jeanne M., Johnston, Bruce A., Ruzich, Constance M., Grant, A. J., “The Challenge of Negation in Searches and Queries," The Review of Business Information Systems 7, 4 (Fall 2003): 63-75. Also in Proceedings of the International Applied Business Research Conference 2003 (Acapulco, Mexico, 2003)

[HAR2003b] Harvey, Valerie J., Harris, Brian, Holdan, E. Gregory, Maxwell, Mark M., and Wood, David F., eds., Discrete Mathematics Applications for Information Systems Professionals (Pearson, 2003). Supplement to Johnsonbaugh, Richard, Discrete Mathematics, 5th ed. (Prentice Hall, 2001).

[HAR2004] Harvey, Valerie J., Wu, Peter Y., and Turchek, John W., “Workshop on Discrete Mathematics for Programs Conforming to ABET IS Accreditation,” ISECON, November 2004, Newport, RI

[HEN2003] Henderson, Peter B., “Mathematical Reasoning in Software Engineering Education,” CACM 49, 9 (September 2003): 45-50.

[HOL2004a] Holdan, E. Gregory, Lias, Allen R., and Maxwell, Mark M., "From Lecturer to Facilitator: The Impact of a Collaborative Effort," International Journal of Learning, 11 (in press).

[HOL2004b] Holdan: E. Gregory, "Getting Closer to a High-Quality Standards-Based Mathematics Classroom: It's not that Difficult! - Issues of Planning, Style, and Change," PCTM Magazine (Winter, 2005; in press).

[JOH2000] Johnson, D. Randolph, “Position Statement.”

[KIL2003] Kilov, Haim, “Discrete Mathematics for Information Systems,” prologue for [HAR2003b], p. v.

[KAR1940] von Kármán, Theodore, and Biot, Maurice A., Mathematical Methods in Engineering: An Introduction to the Mathematical Treatment of Engineering Problems (McGraw-Hill, 1940), p. v.

[LIL2003] Requirements proposal in correspondence from Lauren Lilly, Boise, ID, September 17, 2003.

[PEM2003] Pemmaraju, Sriram and Skiena, Steven, Computational Discrete Mathematics: Combinatorics and Graph Theory with Mathematica (Cambridge, 2003).

[ROD2004] Rodger, Susan, “A Visual and Interactive Automata Theory Course with JFLAP 4.0,” Proceedings of 35th ACM SIGCSE Symposium, Norfolk, VA, 2004, pp. 140-144.

[ROS1997] Rosenstein, Joseph G., Franzblau, Deborah S., Roberts, Fred S., Discrete Mathematics in the Schools (Dimacs Series in Discrete Mathematics and Theoretical Computer Science, Vol 36), (American Mathematical Society, 1997). See also 36.html .

[SCO2004] Correspondence from Elsje Scott, Department of Information Systems, University of Cape Town, Rondebosch, South Africa, November 5, 2004.

[STI1999] Stigler, J. W., and Hiebert, J., The Teaching Gap (Free Press, 1999).

[TIM1995] “Trends in International Mathematics and Science Study,” see Video Studies, Reports/Products at

[WIL2002] Wildstrom, Stephen H., “A Better Web Through Higher Math,” Business Week Online, January 22, 2002.

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download