Some Developments in Mathematical Thinking for Computer Science Education Since Computing Curricula 2001 Doug Baldwin (Moderator)
Bill Marion
Murali Sitaraman
SUNY Geneseo Dept. of Computer Science Geneseo, NY 14454 +1 585-245-5659
Valparaiso University Dept. of Math & Computer Science Valparaiso, IN 46383 +1 219-464-5422
Clemson University School of Computing Clemson, SC 29634-0974 +1 864-656-6738
[email protected]
[email protected] Cinda Heeren
[email protected]
U. Illinois Urbana-Champaign Department of Computer Science Urbana, IL 61801 +1 217-244-2529
[email protected] 10 minute presentations by each panelist, each describing a specific development with which that panelist has been involved, discussing its relationship to CC 2001, and outlining what it bodes for the future
Categories and Subject Descriptors K.3.2. [Computers and Education]: Computer and Information Science Education – computer science education, curriculum. G.2.m [Discrete Mathematics]: Miscellaneous.
30 minutes for discussion between the audience and panel. Suggested topics for discussion include (in addition, of course, to anything that the audience wishes to introduce) the role of mathematics in computer science, whether math’s actual place in computer science curricula has changed since the publication of CC 2001, and if so how. A questionnaire handed out to audience members as they enter the panel session will help them start thinking about these issues.
General Terms Design.
Keywords Mathematics, Computer Science Education, Computing Curricula 2001.
2. Bill Marion
1. SUMMARY
As an outgrowth of the CC2001 report, in the summers of 2003 and 2004, I organized at Valparaiso University two one-week faculty professional enhancement workshops under the auspices of the Mathematical Association of America’s (MAA) PREP project (funded by NSF) for undergraduate mathematics faculty who teach a discrete mathematics course to meet the needs of computer science majors. The purposes were (1) to provide these mathematicians with an understanding of what the implications of the report were for the teaching of discrete mathematics and (2) to illustrate by examples and applications ways to connect the mathematics topics with computer science material their students were learning in CSI and CSII-type courses. Doug Baldwin, Susanna Epp, Peter Henderson, Henry Walker and I were the workshop leaders. One of the outcomes was that, over the course of the next four to five years, some combination of the five of us presented workshops at SIGCSE symposia and CCSC conferences and organized panels at MAA national meetings.
The Computing Curricula 2001 (CC 2001) computer science volume [1] is the first professional society curriculum model for computer science to include elements of discrete mathematics as part of the core of computer science. In the roughly eight years between the release of that curriculum and SIGCSE 2010, computer science education in general has evolved in many ways, and the CC 2001 recommendations have played an influential role in that evolution. This panel reviews developments in integrating discrete mathematics into the computer science curriculum since CC 2001, how those developments have been influenced by the centrality of discrete math in the CC 2001 core, and how the role of math in computer science curricula may continue to develop in the future. The panel will address both the influence CC 2001’s discrete math recommendation has had on curricular changes to computer science courses, and what changes have taken place in the discrete math courses themselves.
More significantly, some of the PREP workshop participants developed a variety of resources for teaching discrete mathematics, including a textbook, proof builder software and an MAA Notes Volume on discrete mathematics.
The panel format will consist of… A 5 minute introduction by the moderator
In my remarks I will summarize these activities and discuss their effect on the teaching of discrete mathematics. In addition, I will
Copyright is held by the author/owner(s). SIGCSE’10, March 10–13, 2010, Milwaukee, Wisconsin, USA. ACM 978-1-60558-885-8/10/03.
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Reasoning tools can be used to introduce one or more mathematical concepts within a week or two in a number of courses, making it possible for educators to experiment with and adapt the ideas without major course restructuring and without significant time investment.
talk about how the CC2001 recommendations have affected the theory and algorithms courses at Valparaiso University.
3. Doug Baldwin The CC 2001 report’s most prominent models for including discrete math in the computer science curriculum are a single course and a two-course sequence. The report’s authors prefer the two-course model as the more workable one pedagogically, but acknowledge that some institutions will not be able to devote two full courses to discrete math. The one-course model is an option for such institutions, although it is not as carefully crafted as the two-course model.
Sitaraman and others have employed mathematical reasoning tools in their classrooms routinely for the past few years at Clemson and elsewhere. Some results from these efforts are reported in recent SIGCSE and ITiCSE proceedings [2]. The feedback from students has been overwhelmingly positive. Students no longer view mathematical thinking as an aside, but as an integral and essential part of computing.
5. Cinda Heeren
In June of 2003, SIGCSE chartered a committee to find or develop practical one-semester model discrete math courses compliant with CC 2001. The committee, co-chaired by Bill Marion and Doug Baldwin, surveyed then-current practice in teaching discrete math. This survey revealed that a large fraction of the respondents did indeed teach discrete math for computer science in a single course. The designs of those courses lay along a spectrum, ranging from courses that were taught with the math itself as the main motivator to courses that were taught with computing applications of the math as the main motivator. As a result of this second observation, the Committee recommended three model courses: two representative of the computer science end of the spectrum, and one representative of the math end.
In 2007, the National Science Foundation supported the creation of a multiple choice learning assessment tool called a Discrete Math Concept Inventory. The development of this tool is occurring over four phases, the first two of which are complete. Our first objective, to winnow the set of inventory topics to a reasonable few, was completed as the result of a Delphi process, in which experts in Discrete Math instruction were asked to name the most important and difficult topics in the course. The consensus response was a tight subset of the CC2001 syllabus, weighted toward proofs and logical reasoning, especially those types most often employed in proofs of algorithm correctness and complexity. The second phase, in which students were interviewed while solving problems from the important-difficult domain, has exposed many specific areas for instructional intervention, particularly in the areas of mathematical rigor and problem solving strategies. While the original purpose of the interviews was to elicit student misconceptions for use in designing multiple choice questions, we have learned much more about how and what students absorb in a CC2001 Discrete Math course. The panel presentation will be a status update on the Concept Inventory development process, together with a guided ride through the analysis techniques used to evaluate student interview responses.
In June 2007 the committee released a final report, detailing the model courses. The committee also released a collection of exercises and syllabi for teaching discrete math. Both the report and collected materials are available on the SIGCSE Web site (see http://www.sigcse.org/resources/discrete-mathematics); the report also appeared in the June 2007 inroads. This presentation will summarize the committee’s history and results, and will seek audience opinions of the impact of the committee’s work on the teaching of discrete math.
4. Murali Sitaraman
6. ACKNOWLEDGMENTS
To teach key mathematical concepts effectively and at the same time engage and excite students in the process, analytical reasoning tools are useful. A variety of CS courses including ones on programming, data structures and algorithms, and objectoriented introduction and software engineering courses offer excellent opportunities to introduce mathematical thinking. A reasoning tool can help introductory students understand why their programs fail and how they can be fixed using mathematical analysis, as opposed to a trial and error alternative. In a course where interface specifications are introduced, students can provide specification-based test points to show their understanding of interface contracts and a tool can check if their test points satisfy given specifications. Alternatively, in a programming or software engineering course, students can study and attempt to prove tool-generated verification conditions for correctness to understand the connection between their code correctness and mathematical thinking.
Thanks to Peter Henderson for help planning this panel.
7. REFERENCES [1] IEEE CS/ACM Joint Task Force on Computing Curricula 2001. Computing Curricula 2001: Computer Science. http://www.computer.org/portal/cms_docs_ieeecs/ieeecs/edu cation/cc2001/cc2001.pdf. [2] Sitaraman, M., Hallstrom, J.O., White, J., Drachova-Strang, S., Harton, H., Leaonard, D., Krone, J., and Pak, R. Engaging students in specification and reasoning: “handson” experimentation and evaluation. In Proc. 14th ACM SIGCSE Conf. on Innovation and Technology in CS Education (ITiCSE), ACM Press, 2009, 50-54.
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