ON ADDING A CRITICAL THINKING MODULE TO A DISCRETE STRUCTURES COURSE *

ON ADDING A CRITICAL THINKING MODULE TO A DISCRETE STRUCTURES COURSE * Michael R. Scheessele 1, Hang Dinh 1, and Mahesh Ananth 2 1 Department of Computer and Information Sciences and 2 Department of Philosophy Indiana University South Bend South Bend, IN 46634 574 520-4815 mscheess@iusb.edu ABSTRACT Coverage of discrete structures is necessary for any computer science curriculum. More broadly, teaching skill in critical thinking is essential to providing a well-rounded college education. We describe our experience with incorporating a critical thinking module (CTM) into a discrete structures (DS) course. INTRODUCTION By designating discrete structures as a Knowledge Area in its 2013 Body of Knowledge, the ACM/IEEE-CS Joint Task Force on Computing Curricula has emphasized the fundamental importance of such topics as formal logic and proof theory to an undergraduate education in computer science [1]. In a similar vein, critical thinking skills are considered intrinsic to a complete college education [2], and these skills are highly valued by prospective employers [3]. Incorporating requirements, such as discrete structures and critical thinking, into a computer science curriculum must occur, however, in a context where other pressures are at work. Some pressures tempt the total number of credit hours required for graduation upward. For example, prospective employers want broadly educated students [3]. Concerns about time to degree, on the other hand, act to hold down the total number of required credit hours [4]. Given these pressures it makes sense to consider connecting related areas of study. Because formal logic, proof theory, and critical thinking are closely * Copyright 2015 by the Consortium for Computing Sciences in Colleges. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the CCSC copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Consortium for Computing Sciences in Colleges. To copy otherwise, or to republish, requires a fee and/or specific permission. 97

JCSC 30, 6 (June 2015) related, our department introduced a critical thinking module (CTM) into our discrete structures course after coverage of formal logic and formal proof techniques. By completing this augmented discrete structures course, our computer science students no longer must take a separate critical thinking course. We describe the circumstances leading to the decision to augment our discrete structures course with a CTM. Then, we describe the new CTM and how we incorporated this into our discrete structures course. Finally, we share observations on the efficacy of augmenting our discrete structures course with a CTM. MOTIVATION By the end of spring semester in 2010, our discrete structures course was in trouble. This course bore the name, Foundations of Digital Computing, carrying a sophomore-level course number, yet computer science (CS) seniors were allowed to take this course in their final semester. This had not always been the case. At one point, this course was a gateway course through which CS majors passed on their way to higher-level CS courses. From the early 2000s on, however, the data structures course served this purpose. Making matters worse, in the late 2000s, the discrete structures course was moved from fall to spring semester. Thus, in principle, a CS student, beginning his or her career in the fall semester and graduating within four years, could take the discrete structures course in the student s final semester. Because the material in this course was foundational, the cart was before the proverbial horse. Complicating matters, the number of topics covered in the discrete structures course was such that instructors reported not having enough time for in-class examples. Something had to be done. By the end of the 2010-2011 academic year, the CS faculty decided to retire the existing discrete structures (DS) course, splitting its content into two new courses. The first new course, Discrete Structures, would be offered relatively early in a student s career and count for two credit hours. The second course, Fundamentals of Computing Theory, would be a senior-level course and count for three credit hours. Thus, while the original course counted for just three credit hours, the two new courses would count for a total of five credit hours. This increase in credit hours would allow more in-class examples. Of course, this would cause the total number of required credit hours to increase by two. Meanwhile, the state legislature was in the process of capping bachelor s degrees offered at state institutions at 120 credit hours [5]. Previously, some in our department had noticed the parallel between content in the old DS course and the requirements for a critical thinking course in the general education curriculum. Although there is obviously much overlap between a DS course, which emphasizes formal logic and proof theory, and a symbolic logic course offered by a philosophy department in support of a general education critical thinking requirement, the latter emphasizes the application of logic to social, political, ethical, and other concerns. Our task then was to develop a CTM that would augment the existing formal logic and formal proof theory material in the new DS course, in order to enhance it to meet the general education committee s criteria for a critical thinking course. This new course would simultaneously satisfy a CS student s DS and critical thinking requirements, preventing the total number of required credit hours from increasing. 98

CCSC: Northeastern Conference CRITICAL THINKING MODULE (CTM) In this section, we first give the criteria which our module had to meet, in order for the general education committee to approve our DS course as also satisfying the critical thinking course requirement. Then, we describe how our proposed module met these criteria and how this module was implemented. General Education Requirements for a Critical Thinking Course Adding the CTM required that our new two-credit Discrete Structures course be worth three credits instead. Because the critical thinking courses already offered were three-credit courses, and assuming the general education committee approved our proposal, there would be no net change in the total number of credit hours resulting from the split of our old DS course into two new courses. The general education committee at our institution requires that the following five criteria be met, in order for a course to count toward the critical thinking requirement [6]. In particular, such a course must: 1. provide instruction in identifying and differentiating questions, problems, and arguments; 2. teach students how to evaluate the appropriateness of various methods of reasoning and verification; 3. teach students how to identify and assess stated and unstated assumptions, and critically compare different points of view; 4. introduce techniques for evaluating the quality of evidence and reasoning; 5. require students to formulate questions and problems, construct and develop cogent arguments, and articulate reasoned judgments. CTM Design The overall design of the CTM focused on satisfying the criteria listed above. In this section, we outline the overall module design. We begin, though, by explaining how the module was incorporated into the DS course. To incorporate the CTM into the DS course, we proposed allotting four or five 75-minute class periods, accompanied by several homework assignments. We also proposed that this module be introduced after the normal DS material covering formal logic and formal proofs. By covering this prior to introduction of the CTM, students would have knowledge of and practice with the concepts of statements, logical connectives, derivation rules, structure and components of an argument, deductive reasoning and some of its common patterns (e.g. modus ponens, modus tollens), as well as the notion of argument validity. Then, the critical thinking instruction would re-orient this knowledge and experience to the practice of logic and reasoning beyond the domains of science and mathematics. Regarding criterion 1, by just using typical DS material, students are instructed in how to identify and differentiate questions, problems, and arguments. Beyond the domains of science and mathematics, however, we needed to propose to the general education committee some instruction in how to differentiate questions, problems, and arguments. To address this, we proposed that the first lecture of the CTM re-introduce logic and argumentation from the perspective of how this material is typically covered at the start of a philosophy course. Because our students would already have experience with logic and argumentation (e.g. proof), coverage could be brief, concentrating on what 99

JCSC 30, 6 (June 2015) is pertinent to critical thinking. For example, the issue of a deductive argument s soundness requires more instruction in the context of critical thinking. Also, reasoning in an introductory DS course will be almost entirely deductive, whereas inductive reasoning is sometimes necessary in the context of critical thinking. Regarding criterion 2, in a typical DS course, students are taught how to evaluate the appropriateness of various methods of reasoning and verification in the context of science and mathematics. This is accomplished by covering common patterns of deductive reasoning, such as modus ponens and modus tollens. In addition, in the context of informal proofs, a variety of techniques are covered: disproof by counterexample, direct proof, proof by contrapositive, proof by contradiction, and proof by mathematical induction. Beyond the domains of science and mathematics, we proposed that the first lecture to re-introduce logic and argumentation from a critical thinking perspective would satisfy criterion 2, because after students are taught to identify arguments and distinguish between deductive and inductive arguments, they would be taught how to determine whether a deductive argument is valid and sound and whether an inductive argument is strong and cogent. Satisfying criteria 3 and 4 required more effort. In an early-career, introductory DS course, the issues of unstated assumptions, comparing different points of view, and evaluating the quality of evidence are not likely to feature as prominently as in an introductory critical thinking course. In order to address criteria 3 and 4, we proposed two lectures to cover the broad topic of critical analysis. Outside the domains of science and mathematics, it can be challenging to identify and evaluate the argument of an author or speaker. Regarding difficulties in identification, there may be several sources. For example, nonargumentative prose may be interspersed throughout an argument [7]. In addition, arguments may be extended, where the conclusion of an argument serves as the premise of a subsequent argument [8]. (Extended arguments or proofs are less emphasized in an early-career, introductory discrete structures course.) With respect to challenges in evaluation of an argument, there also may be several sources of difficulty. For example, unstated assumptions make critical analysis difficult. Although arguments and proofs in computer science and mathematics can have unstated assumptions, those working in a subdomain likely share these assumptions. This may not be the case in domains outside science and mathematics, where cultural, social, and personal factors may provide a larger part of the background context. Additionally, while definitions in science and mathematics are typically agreed upon, this may not be the case in other domains. Yet another challenge is evaluation of the evidence given in support of a claim. Again, coverage of this will probably be more prominent in an introductory critical thinking course. Finally, when critical thinking is applied in construction or analysis of an argument, sometimes a formal approach is warranted. In dynamic contexts, such as verbal presentations, debates, and ordinary conversations however, treating arguments as if they were proofs to be constructed or reconstructed and analyzed may not be feasible. Thus, we added to our module material on how to detect bad arguments. The basic strategy is to check for faulty premises and to check for common fallacies in reasoning [7]. Regarding criterion 5, we proposed using a fourth class period to examine a case study, in order to give students hands-on in-class practice with critical thinking. By 100

CCSC: Northeastern Conference proposing to incorporate this CTM into our DS course, the general education committee approved the course as a critical thinking course. CTM Implementation In order to implement both the re-introduction of logic/argumentation from a critical thinking perspective and the instruction in critical analysis, we developed a document, Logic and Critical Thinking, and a corresponding set of PowerPoint slides. The document, which serves as the text for the CTM, is based on an earlier document developed by the third author as a quick tutorial for learning to do analytic philosophy in higher-level philosophy courses. Later, the first author adapted that document for use in an introductory cognitive science course. Thus, the nucleus of the current document has been used successfully in a number of courses. The current document incorporates more topics, description, explanation, and examples than the original document because the scope of instruction of the CTM is wider than that of the philosophy and cognitive science courses. Even so, we tried to remain true to the spirit of the original document. The document is divided into five sections. Section I, Introduction, re-introduces logic and argumentation to DS students who have already studied formal logic and formal proofs. Section II, Two Basic Types of Argument, covers how to distinguish deductive from inductive arguments. For deductive arguments, students are shown the difference between valid and invalid deductive arguments. For valid deductive arguments, students are shown the difference between sound and unsound arguments. For inductive arguments, students are shown the difference between strong and weak inductive arguments. For strong inductive arguments, students are shown the difference between arguments which are cogent and not cogent. Section III, Critical Analysis, describes the steps involved in identifying and evaluating arguments. This section also covers issues which make critical analysis difficult, including the possibility of unstated assumptions and the challenge posed by definitions. Section IV, Good versus Bad Arguments, reminds students of common deductive reasoning patterns, such as modus ponens and modus tollens. This section also covers the importance of evaluating premises and evidence, as well as common logical fallacies. Finally, Section V includes several references containing material on critical thinking. The document, accompanying PowerPoint slides, and several homework assignments can be found at http://1drv.ms/1t5s1rr. For the in-class case study, in spring 2014 we used the vaccination-autism controversy. Materials can be found at: http://www.annenbergclassroom.org/page/the-battle-of-the-experts. In fall 2014, we used President Obama s 2014 State of Union for the in-class case study. This case study was adapted from an exercise named Facts of the Union also from www.annenbergclassroom.org. CONCLUSIONS The first author of this paper taught the new DS course for the first time in spring 2014. The second author taught the course in fall 2014. Formal assessment of the new 101

JCSC 30, 6 (June 2015) CTM would not be very meaningful yet due to issues related to retiring the old DS course and introducing the new one. Specifically, students enrolled in the new course in spring 2014 still mostly had either junior or senior standing, although we are ultimately targeting sophomores and first-semester juniors. Even so, we can offer some preliminary observations on the effectiveness of the CTM. First, students were able to grasp the basic mechanics of identifying and evaluating arguments. This was expected because students had just been taught formal logic, formal proofs, and informal proof techniques, prior to seeing the critical thinking material. On the other hand, students found the critical analysis of real text to be more difficult. Many students struggled to identify and evaluate an argument taken from a real text. Forcing students to do real work in critical analysis seems necessary, even when competence in critical thinking on simpler examples is evident. We are also interested in exploring how the incorporation of the CTM could help us teach traditional topics of DS more efficiently. In our DS course, the most challenging topic seems to be informal proofs. It is typically covered immediately after formal proofs. While students have no major problems with formal proofs, many struggle with informal proofs. One reason could be a lack of smooth transition from formal to informal proofs. We envision that the CTM may address this challenge. Observe that critical thinking and informal proofs share skills not covered in formal proofs. One such skill is identifying unstated assumptions. In formal proofs, every assumption must be added to the proof sequence explicitly. In informal proofs, however, arguments with implicitly assumed axioms are common. When moving directly from formal to informal proofs, students may get confused if not instructed in how to identify unstated assumptions. It may be gentler to introduce such skills in general contexts with which students are comfortable. As such, the CTM could fill the skills gap between formal and informal proofs. We conjecture that teaching critical thinking skills may improve students grasp of informal proofs. Our plan is to test this conjecture in our new course by observing whether (Fall 2014) or not (Fall 2015 or later) covering the CTM before informal proofs improves performance on informal proofs. REFERENCES [1] The Joint Task Force on Computing Curricula: ACM/IEEE-CS, Computer Science curricula 2013: curriculum guidelines for undergraduate degree programs in Computer Science, December 2013, http://www.acm.org/education/cs2013-final-report.pdf, retrieved October 22, 2014. [2] Foundation for Critical Thinking, The Critical Thinking Community website, http://www.criticalthinking.org/pages/college-and-university-faculty/798, retrieved October 22, 2014. [3] Association of American Colleges and Universities, Hart Research Associates, It takes more than a major: employer priorities for college learning and student success, 2013, http://www.aacu.org/leap/documents/2013_employersurvey.pdf, retrieved October 22, 2014. 102

CCSC: Northeastern Conference [4] Johnson, N., Reidy, L., Droll, M., LeMon, R. E., Program requirements for associate s and bachelor s degrees: a national survey, 2012, http://completecollege.org/docs/program%20requirements%20-%20a%20natio nal%20survey.pdf, retrieved October 22, 2014. [5] Indiana Commission for Higher Education, New state laws support college completion, student success, 2012, http://www.in.gov/che/files/120312_release_-_legislative_recap.pdf, retrieved October 22, 2014. [6] IUSB Task Force on General Education, Report and recommendations, March 2003, https://www.iusb.edu/general-educ/course-approval-docs/general-educ-critical-th inking-characteristics.doc.php, retrieved October 22, 2014. [7] Vaughn, L., Doing Ethics: Moral Reasoning and Contemporary Issues, 2 nd Edition, New York: W.W. Norton & Company, 2010. [8] Bowell, T., Kemp, G., Critical Thinking: a Concise Guide, New York: Routledge, 2002. 103