A first course in discrete structures with applications to computer science

Discrete Structures texts are notorious for their lack of examples from computer science. One response to this problem is to teach the Discrete Structures course as a course in pure mathematics. However the end result is computer science students completing the course without understanding the reason it is required. A second response to the problem is to give the course a computer science flavor by requiring the writing of programs illustrating the mathematical ideas introduced. This approach still does not address the real issue of why the course is required. A third response, the one advocated by this paper, is to supplement the text with examples and exercises showing how the mathematical concepts are actually applied in various areas of computer science. This paper examines some appropriate computer science examples and exercises and discusses the author's experiences when incorporating them into the Discrete Structures course at S.U.N.Y. at Plattsburgh.

It is generally accepted that a course in discrete structures such as Course B3 in Curriculum 68 recommendations [1] is an important part of a Computer Science program. In many institutions a Discrete Structures course is offered in the Computer Science Department although it overlaps courses in the Mathematics Department [2,3,4]. In smaller schools the size of the student body and staff limitations make such a solution impractical. At Augusta College we have met the objective of early exposure to discrete mathematics with applications in computer science by offering a two-hour computer science course as a co-requisite to a three-hour mathematics course. While this approach is a compromise it does appear to have some merit.

It is generally accepted that a course in discrete structures such as Course B3 in Curriculum 68 recommendations [1] is an important part of a Computer Science program. In many institutions a Discrete Structures course is offered in the Computer Science Department although it overlaps courses in the Mathematics Department [2,3,4]. In smaller schools the size of the student body and staff limitations make such a solution impractical. At Augusta College we have met the objective of early exposure to discrete mathematics with applications in computer science by offering a two-hour computer science course as a co-requisite to a three-hour mathematics course. While this approach is a compromise it does appear to have some merit.

Over 350 computer science students take Russ Miller's Discrete Structures course every fall semester. About 90 percent are freshmen. By week five, they are breaking problems into small chunks and learning ways to solve each chunk at the same time-in parallel. Today, multicore processors power our laptops and cellphones. Distributed cloud servers or supercomputer clusters process large data sets to improve Facebook news feeds or predict the weather. To take full advantage of these systems, you need parallel algorithms. "It's a parallel world," says Miller, a computer scientist at the State University of New York at Buffalo. "Why is no one teaching a course in parallel algorithms to freshmen?"

In a previous study, we investigated a framework for teaching abstraction in computer science (CS) in the context of an introductory CS course for 7th-grade students. The framework was found effective for developing CS abstraction skills. In this paper, we describe a study that investigated teaching abstraction in CS to 7th-grade students through the perspective of gender. Specifically, we compared the CS abstraction abilities of boys and girls taking the introductory CS course in 7th grade (using the Scratch environment). We also examined the difference in the effect of the teaching strategy on CS abstraction abilities between boys and girls. Our findings show a significant advantage of girls over boys regarding several aspects of CS abstraction. We also found that the framework for teaching CS abstraction, which was already established in our previous study as effective in the context of an introductory course for 7th-grade students, has an enhanced effect in this context when it comes to girls. This enhancement was manifested by a better overall performance by the girls, when comparing boys and girls that took an introductory CS course in which this framework was used. It was also manifested by the girls' deeper and more reliable perception of CS. We conjecture that this framework has the potential to affect girls' self-efficacy, and as a result to create a positive effect by encouraging and motivating girls to pursue CS education beyond this introductory course. That is, this framework has the potential to contribute to solving the issue of underrepresentation of females at the different stages of CS education, or in other words, to widening the CS pipeline for females.

After years of discussion, the consensus among academic computer scientists seems to be that undergraduate computer science programs need more mathematical content. This mathematical content is not to be found in areas such as calculus and differential equation but in areas such as discrete mathematics and mathematical logic. The objective of this article is to re‐examine the central role of mathematics in computer science by reviewing the undergraduate computer science curriculum from Curricula ‘68 to the current Computing Curricula 1991. The trend toward a greater concentration in such courses as discrete mathematics will be explored. Furthermore, we collected and put together comments and concerns of three prominent computer scientists about the importance of mathematics in computing science and lack of mathematical content in computer science programs. We then look at one area of computer science amenable to mathematical treatment. This area, software specification, is one of the important phases of the software development. The use of formal methods based on discrete mathematics has been increasingly emphasized recently for software specifications in an effort to produce precise and concise descriptions. We will provide an example to demonstrate the importance of a mathematical approach to arrive at precise specifications and argue against claims that mathematical notations introduce unnecessary complication by pointing out the advantages of and misconceptions about the use of formal notations. The article is concluded by once again emphasizing the integration and applications of mathematics in all areas of computing discipline, especially for software development.

A one-semester introductory undergraduate course in computer game development is described. The freshman-level course has been taught for five years. The course was designed and implemented in the context of a new curriculum in computer game development. Goals of the course were to retain undergraduate computer science (CS) majors, attract new students to CS, and expose students to the content creation pipeline used in advanced courses. A significant aspect of the course is the end-of-semester project to create a machinima - a video created using computer game technology. Besides the initial goals for the course, other benefits were discovered including fostering a sense of community. Results, both quantitative and anecdotal, show the course has exceeded expectations in meeting the goals originally established. The course has been integral to the growth of the CS program, enabling the program to nearly triple in size over four years.

As undergraduate computer science enrollments continue to grow, individualized instructor attention becomes increasingly scarce. The impact of social distance between students and their teachers is particularly apparent in large introductory classes, exacerbated by students' lack of common prior experience in computer science. Some institutions remedy class size and experience gaps by hiring advanced undergraduate students as teaching assistants for their introductory courses. However, without the resources to carefully hire, train, and manage undergraduate teaching assistants (uTAs) during the semester, their potential as trustworthy peer mentors and helpful tutors often goes unrealized. This poster presents details of the uTA hiring process, training course, and management strategies used during the 2022-2023 academic year for the introductory computer science courses at the University of Delaware (UD), a large, public, research-focused institution. This system was designed for introductory CS classes, with the goals of low instructor overhead, long-term scalability, and the development of empathetic teaching assistants who could motivate students toward successful learning. Highlights of the system include a hiring process that considers candidates' personality and enthusiasm alongside their technical skills, asynchronous training provided through the university's learning management system, and the use of a head uTA to manage other staff members. Following the implementation of these policies, instructors and uTAs alike reported positive experiences compared to previous semesters without significant changes to students' learning outcomes.

Graph theory is an integral part of a discrete structures course required in computer science programs. Unfortunately, in the short time such courses devote to graph theory, students often do not appreciate and/or recognize how graph theory notions and theorems become problem modeling and problem solving tools. Graph Works is a visualization tool to show what properties of graph mean in terms of a particular graph and what properties and substructures fundamental theorems guarantee to exist in a graph. The tool is designed to help users understand and develop algorithms that use graph properties needed for problem solving or find and use particular subgraphs as part of the solution process.

The proportion of computer science (CS) bachelor's degree recipients who are women has consistently been small and is declining. This study investigates factors that predict performance and persistence in an undergraduate CS program and explores why even high-achieving students leave the undergraduate "CS pipeline." The factors that predict achievement and retention sometimes interact in complex, unexpected ways. Male students who earned less than a B in an introductory CS course were more likely to take the next course in the curriculum than were women who earned less than a B. Achievement is a factor in even high-achieving students' decision to leave CS; loss of interest can accompany loss of confidence. Level of achievement was predicted by various background factors including Scholastic Aptitude Test scores, the number of Calculus courses taken before entering the CS program, amount of access to a computer at home, prior computing experience, and having a mentor or role model during high school. Most of these factors also predicted persistence beyond the first two courses required for a CS major. Curiously, women in the introductory CS course who reported having low exposure to specific programming skills outperformed women who reported having a high level of programming experience. The reverse was true of men. Further investigation provided evidence that women who develop programming skills while in high school might do so at the expense of developing other skills that strongly predict CS achievement, particularly math skills.

We are concerned about a view in undergraduate computer science education, especially in the early courses, that it's okay to be math-phobic and still prepare oneself to become a computer scientist. Our view is the contrary: that any serious study of computer science requires students to achieve mathematical maturity (especially in discrete mathematics) early in their undergraduate studies, thus becoming well-prepared to integrate mathematical ideas, notations, and methodologies throughout their study of computer science. A major curricular implication of this theme is that the prerequisite expectations and conceptual level of the first discrete mathematics course should be the same as it is for the first calculus course --- secondary school pre-calculus and trigonometry. Ultimately, calculus, linear algebra, and statistics are also essential for computer science majors, but none should occur earlier than discrete mathematics. This paper explains our concerns and outlines our response as a series of examples and recommendations for future action.

Computer Science educators are constantly reinventing introductory CS0, CS1 and CS2 courses to retain students and increase their learning and motivation. While the focus on drawing students into Computer Science is vital to maintain enrollment, it is also paramount to sustain student motivation by using new pedagogical approaches, contextualized to how students learn, throughout the curriculum to develop technical and interpersonal skills. Many educators have developed one-off courses employing new technology or a project-driven approach to engage students and transfer ready industry skills. Yet, too few have woven a combination of pedagogical approaches sensitive to how students learn with in-demand technology and skills contiguously throughout the upper-level curriculum and tied it to opportunities outside of the classroom. In this poster, we present the initial results of a project with two aims to: (1) design, pilot, and offer new curriculum materials for three upper-level Computer Science courses using a mobile computing platform paired with pedagogical approaches sensitive to the learning styles of today's student; and, (2) develop out-of-classroom learning opportunities including mobile computing student seminars, externship opportunities and service-learning projects reinforcing in-classroom technical skills. We believe that these two components synergistically, can be leveraged to revitalize the undergraduate Computer Science curriculum, better motivate student learning and provide in-demand technical skills to students without losing the fundamental Computer Science concepts. This project is partially supported by NSF DUE#1140781.

Introductory computer science courses offer an excellent test bed for utilizing a personalized system of instruction. The ubiquity and operating hours of computer terminals and computer centers in educational institutions is ideal for student self-pacing. The rapid response time of computers is a positive reinforcement when solving problems. The recent advent of the PLATO computer-based educational system provides many opportunities for students and faculty in computer science and many other disciplines. PLATO terminals provide a powerful motivational tool for PSI. Motivation is a critical part of computer science courses. Students are expected to acquire synthesis-level skills in the cognitive domain. This level of knowledge is higher than the level of skill, memorization or rule application, required for many other introductory courses. An Excellent Mixture for PSI: Computer Science, PLATO, and Knowledge Levels.

There has been much debate in the past few years about the appropriate mathematics requirements for an undergraduate computer science major. The discussion has focused primarily on two issues: (1) the underlying mathematical content of computer science courses and (2) the content of mathematics courses which would serve as appropriate cognate requirements for computer science major programs. While this discussion has been helpful, it has been too narrowly focused--it has not started from an understanding of the relationship between the disciplines of mathematics and computer science, but rather has sought to identify mathematical prerequisites that computer science majors need in order to take existing computer science courses. This paper is a small step in seeking to apply an understanding of the relationship between the disciplines of mathematics and computer science to the undergraduate computer science curriculum.

Most studies of gender and computer science have been conducted (out of necessity) in gender-imbalanced environments. The findings often point to significant gender differences, leading the researchers to recommend strategies to meet these differences. One such recommendation is to adopt a female-friendly, contextual approach to curriculum development.1 In contrast, based on our recent and ongoing studies of undergraduate computer science (CS) students at Carnegie Mellon University, we hypothesize that in a more balanced environment, gender differences tend to dissolve. That is, the spectrum of interests, motivation, and personality types of men and of women becomes more alike than different. This leads us to be considerably more pragmatic in our recommendations for effective and positive change. Indeed, we believe that recommendations for curricular changes based on presumed gender differences can be misguided and may help reinforce, even perpetuate, stereotypes. Here, we describe some of the changes in the culture of computing as our undergraduate CS environment has become more balanced in three critical domains: gender, the mix of students and breadth of their interests, and the professional experiences afforded all students. Brief Background for the Evolving Culture In 1995, just 7percent (7 out of 96) of the entering freshmen CS majors at Carnegie Mellon were women. Since 1999, the percentage has increased nearly fivefold (on average, 45 out of 132), challenging trends across the country.2 Two major factors have contributed to the dramatic increase of women in CS at Carnegie Mellon: an outreach program for high school CS teachers that combined technical training with discussions of gender-gap issues and a broadened admissions policy (emphasizing diverse interests—along with high [End Page 110] achievement in mathematics and science—and de-emphasizing prior programming experience). To meet the needs of students with varying backgrounds, multiple entry routes were built into the first-year programming sequence.3 The subsequent creation of the proactive student organization Women@SCS has been catalytic in building an environment in which the new student body can flourish.4 Women@SCS explicitly provides crucial educational and professional experiences generally taken for granted by the majority in the community but typically unavailable for the minority participants.5 Many of these experiences are casual and often happen in social settings. For example, in an undergraduate CS program, male students often have the opportunity to discuss homework with roommates, with friends late at night, or over meals. Course and job information and recommendations are passed down from upperclassmen, from fraternity files, or from friends. Women students, being in the minority, do not have access to—in fact are often excluded from—these implicit and important advantages. As one proceeds into the professional world, similar phenomena occur. These key actions, and implications for other venues, are discussed further at the end of this paper. Our recent study, based on a set of interviews with seniors in the class of 2002, may be viewed to a limited degree as a follow-up to the intensive longitudinal studies of CS students at Carnegie Mellon carried out by Jane Margolis and Allan Fisher between 1995 and 1999.6 Their studies, funded by the Sloan Foundation and discussed in Unlocking the Clubhouse, were undertaken to help understand differences in male and female students' engagement (attachment, persistence, and detachment) with computer science, with a special focus on the gender imbalance in the field.7 A major goal was to devise and effect changes in recruitment, curriculum, pedagogy, and culture to encourage the broadest possible participation in the computing enterprise. Unlocking the Clubhouse paints a bleak picture for women in the undergraduate computer science program in the 1990s, one that still resonates with many CS programs across the nation. Margolis and Fisher note that for the most part, "women who were enthusiastic about computing find their confidence and interest extinguished in the college years."8 Consequently, many of the (few) women students who entered the...