Using an On-Line Platform to Teach Logic 1
Leonel V. Morales Díaz and Moris A. Polanco 1 2
Universidad Francisco Marroquín/Guatemala, Guatemala Universidad Francisco Marroquín/Guatemala, Guatemala
Abstract—Starting in January 2009, the School of Economic Sciences at the University Francisco Marroquín has been using the Open Learning Initiative—OLI—platform from Carnegie Mellon University to teach the courses Symbolic Logic and Logic and Critical Thinking. These three years of experience have yielded important lessons for teachers as well as for students, faculty and platform administrators. In this short time the way the course is being taught and even the methods of evaluation have evolved, proving the flexibility both of the technology and the attitude of the stakeholders towards it. We often think that teachers and students are ready for adopting new methods in education but these changes may prove tricky to embrace. This report may help in the process. Index Terms—Logic, instruction.
I. INTRODUCTION When the Francisco Marroquín University—UFM— was planning courses and contents for the first year of its new program “Entrepreneurial Engineering” at the School of Economic Sciences, an effort was made to identify an adequate on-line platform for teaching the course whose content had to do primarily with symbolic logic. The Open Learning Initiative—OLI—from Carnegie Mellon University and its course Logic & Proofs seemed suitable for the task. The program opened in January 2009; first-year students started classes using the platform as a repository for course material, concept-clarification exercises, practice problems, laboratories and eventually for mid-term and final exams. Choosing a technology outside the university was not the only path in this case. UFM had—and has—its own portal, known as MiU, for course scheduling, document exchanging, homeworks reception and grades reporting that in fact is used by most teachers and students. However, this portal is only capable of holding “static” contents—uploaded by teachers, downloaded by students —or receiving work done by students outside the portal and then sent to the teacher like an email or a blog post. This technology was not suitable for the goals in the design of the course and, somehow, appeared outdated and out of focus. The decision was then to go with OLI and Logic & Proofs. The course design for Symbolic Logic was very similar to a course in the Business Administration program so both Symbolic Logic—the course for Entrepreneurial Engineering—and Logic and Critical Thinking—for Business Administration—ended up using almost the same on-line material at the same time that first semester of 2009.
At the time of preparing this paper the courses have been taught three times to more than 500 different students by at least 5 teachers, in close coordination and frequent communication with platform and content managers at Carnegie Mellon University. A considerable amount of experience has been gathered by the faculty members involved. Our goal in this work is to present what we think may be key factors and important advice to bear in mind by anyone considering teaching a course with a web-based tool, particularly those courses related to logic, Boolean algebra, discrete mathematics and similar subjects. The paper starts with a short description of the three elements involved: a) the OLI platform, b) the Logic & Proofs course, and c) its incarnation in the Francisco Marroquín University: the Symbolic Logic and Logic and Critical Thinking courses. These descriptions are provided in Section I. In section II we summarize our experience and try to identify the key factors that seem to improve the probability of success in the adoption of an on-line platform particularly for the case of a symbolic logicrelated subject. Conclusions are then presented in section III along with our plans for future work in section IV. A. The OLI Platform The Open Learning Initiative—OLI—is an open educational resources project at Carnegie Mellon University that began in 2002 with a grant from The William and Flora Hewlett Foundation . Most of its courses are designed so that students can learn at their own pace without an instructor. For formal courses being taught as part of a higher education program, the teacher uses the material to support face-to-face classes, let the students solve problems as homeworks and even take exams. Teachers have plenty of resources to monitor and control the progress of the students. Some are very clever, for example the Digital Dashboard for Learning  that collects information as students complete work in the class, and the Gradebook that helps teachers and assistants to grade individual students. An interesting feature of the OLI project is that anyone can register on its web site and enroll in one of the many free courses available, that include subjects such as French, Physics, Biochemistry, Visual Communication Design, Empirical Research Methods, Economics, among others. Even if a visitor does not want to register, there is an option to “peek-in” course contents and see the available material.
Free courses do not provide instructor guidance or any form of human help and they won't be awarded credits by Carnegie Mellon or any other educational institution as far as we know. The OLI web site can be visited at: http://oli.web.cmu.edu/openlearning/index.php. B. The Logic & Proofs Course The Logic & Proofs course is part of the OLI project and is presented as an introduction to modern symbolic logic very much in line with what is taught in courses like mathematical logic, logical structures, Boolean algebra, discrete mathematics, logic circuits and others. The course is divided into two main parts: sentential logic—also known as propositional logic— and predicate logic where predicates, individual constants, variables and universal and existential quantifiers are explained. However, the design of the course, the exercises, the problems to solve, and many other resources in it, emphasize what is called strategic argumentation, primarily the work of Wilfried Sieg , in which a set of carefully structured moves is used to guide—algorithmically—the resolution of any derivation once the goal—the logical formula that has to be demonstrated—is known. The discussion of strategic argumentation is well beyond the scope of this paper. The interested reader will find a good starting point to understand its nature in .
A symbolic logic course without problems to solve would be quite theoretical and would hardly achieve its pedagogical objectives. Because of that Logic & Proofs provides a huge number of very well crafted exercises and solving tools designed to be used on-line. Many of them are embedded in course readings so students can test what they have learned immediately after reading the text. Others require the use of the TruthLab, see Fig. 1—for tasks like building a parse tree to check the syntactical correctness of formulas, evaluating their truth value given a truth assignment to atomic sentences, constructing truth tables or validating formulas by using truth trees—and for derivations—where most of the work of students is made in the course—the ProofLab. Both of them are Java based and will only run in Java-enabled browsers, which does not seem to be a problem nowadays. The ProofLab, Fig. 2, is the most sophisticated of the tools available in the course. Its construction is based on the strategic argumentation procedures, and, as we will point, is highly effective for learning. It includes the ProofTutor, a highly sophisticated interactive tool that provides intelligent advice to students. When a user is stuck on a proof the ProofTutor is able to provide hints built dynamically from the internal proof generation engine called AProS: beginning with the student’s partial proof, AProS constructs a complete proof which the tutor analyzes and transforms into a series of hints that can be
Figure 1. The TruthLab allows students to build truth tables and truth trees like the one shown here. Truth trees are used to determine if an argument is valid or find a counterexample if it is not.
Figure 2. The ProofLab is where students practice logic proof construction. The ProofTutor at the bottom right pane, can provide guidance and hints for constructing each proof.
passed to the student. The hints are given in order from the most general strategic ones to the more specific and concrete. The last hint in the sequence is a recommendation for a particular step in the proof construction . Students usually try first to construct the proof by themselves and then turn to the tutor for advice when needed. All the tools provide feedback to the student and report back to the teachers so all the activity of the students can be graded. C. The Symbolic Logic and Logic and Critical Thinking Courses Symbolic Logic is the name of the course whose content and evaluation is highly supported by Logic & Proofs and the OLI platform. It is part of the Entrepreneurial Engineering program at the School of Economic Sciences at Francisco MarroquĂn University. The course is taught to first year students in the first semester along with calculus, physics, and courses related to business management and administration. In the design of the programs symbolic logic was preferred to a logic course in line of Aristotelian logic, because it better supports the type of problems that engineers face in their education or their professional career. Computer programming, digital circuit design, data base querying, industrial design, as well as several topics in engineering are highly symbolic and highly logical. After three years of existence of the program, the course has proved useful and consistent with these considerations.
There is basically no course content outside of that provided by Logic & Proofs, although we are considering including in the future some topics on digital circuitsâ€”as a practical applicationâ€”or Prolog programming. The course duration allows teachers to devote two periods of 80 minutes a week to each of the chapters of Logic & Proofs with only two weeks of class left that are used for reviewing past content. Logic and Critical Thinking is a course in the Business Administration program that is offered to first year students in the first semester with the same duration and the same amount of weekly classes as Symbolic Logic. When the Entrepreneurial Engineering program started, both courses adopted the OLI platform and the Logic & Proofs material but Logic and Critical Thinking covered only the first part: sentential logic, leaving predicate logic out of the syllabus. After a couple of years, seeing the positive results of going through the whole course, the syllabus was changed and at the moment of the preparation of this paper both courses almost cover the same material with subtle differences in the way it is presented to students. For engineering students the emphasis is made in symbolization of problems, operation and the understanding of rules. For business administration the reasoning pattern presented by each logical rule is especially reviewed. Students are required to enroll in the paid version of the course, not in the free one, because many of the benefits of the platform are not available in the latter, including tracking of activities and problems, homeworks and test scoring.
The use of OLI was not intended to replace face-to-face instruction in the classroom but to serve as the supporting platform for the course. II.
A. Class Preparation Although OLI courses are designed to be self-taught, it is not recommended to rely on student reading and understanding the material in advance. A good and motivating face-to-face class is also needed. An important portion of the responsibility of the teacher will be to introduce the platform to students and showing them how to navigate contents, where the exercises are located, what is expected from them to accomplish every week and— highly important—how their work is going to be reflected in the Gradebook. The class topic can be presented using traditional resources like blackboards, projectors, presentations, solved problems, etc., but the use of the on-line tools must not be left out, see Fig. 3. A session in the laboratory is highly recommended with one computer per student—if possible—to see them in action using the tools. All sorts of issues can be spotted and solved during these sessions including people not knowing how to browse to the web page, lack of username (unregistered students), browser incompatibilities, content navigation, etc.
Figure 3. “Did I Get This?” exercises are embedded in the text of the course and can be used as in-class activities.
B. Getting Students Involved Homeworks that have to be solved on-line, inside the platform, with a deadline are highly important. They will force reluctant students into the web and will allow them to taste the real experience of using the tools by their own means or with the help of other students or teachers. As the students see their work graded and their advance reported they will feel motivated to get involved in the class in an active way. OLI and Logic & Proofs provide constant feedback, hints, scores and percentage of advance reports to students , that gratifies the work done and keeps students longing for more. To facilitate the process each student must be allowed to develop a personal way of working in the platform, for instance by meeting with friends to work out exercises and problems or going alone and turning to teachers to solve doubts personally or through email.
C. Communication Social networks are a popular way of exchanging messages between small or big groups and their integration into the platform—or the platform into the social network—is highly recommendable but not indispensable. The real requisite is to have a wellestablished communication channel that is suitable, trusted and comfortable for most of the students. It can be email, a blog, a Twitter account, or a more traditional one like office hours for class review and question answering. One teacher of Logic and Critical Thinking created a Facebook page (http://facebook.com/logicasimbolica) to facilitate the communication among students; it has now 189 followers. Some students will need special attention and will try to find a way to get it. For example, some will come to see the teacher or assistants with their portable computers to showcase particular situations while others may do the same through email attaching a screen-capture of the issue. D. Evaluation The way the course work is to be evaluated and the percentage of the grade for each homework or test must be defined at the beginning of the course—as it is done traditionally—but the teacher and the students should be prepared to implement an innovative way to evaluate contents. In the case presented here, we have usually mixed the types of evaluations. For example, for the first mid-term exam the test was handwritten—and students protested that they were doing a lot of work on the computer but were being evaluated per work on paper—and then switched to an on-line test for the second mid-term or final exam—and then comments from students regarded how difficult or extensive the test was. Some teachers have devised ways to evaluate partially in paper and partially on-line for the same mid-term exam. E. Platform and Contents Support The OLI platform has been in service for almost 10 years now for a wide range of courses and has a responsive and efficient team that is ready to help and trouble-shoot problems as they appear. Their technology is backed by research not only for the platform built but for the methodologies it helps to implement . On top of that, course contents are also researchsupported  and the researchers work closely with platform supporters to guarantee the delivery of quality processes. In our case we added good coordination practices among the staff of teachers, with almost weekly meetings to share experiences and plan ahead of class development. Several times we turned to the content designers or to platform developers to seek support, help or advice in the way we were using the technology and found them accessible and prompt to answer. F. Key Factors A careful examination of the experience gathered reveals that one key factor for the success of the adoption of the on-line platform is the availability of a huge number of ready-made activities—exercises, problems, reviewing questions, tests, etc. —that turn the platform into a living,
dynamic and attractive workplace. If the web is only composed of static contents – thought of as a repository of texts, graphics and explanations—then it will not effectively support the teaching process. This huge amount of interactive content, if properly organized, can support classroom teaching, homeworks, tests, exams and even extra-points activities. Specialized tools—like the TruthLab and the ProofLab mentioned earlier—provide capabilities that would be very hard to match with standard-generic platforms. From the viewpoint of the student, obtaining immediate feedback in the form of scores, comments or even percentage of work done with the gratification it provides, is key to get them involved in the web platform and acts as a motivation to keep coming back for more. It is also important to point to the type of problems that the course confronts students with. At first—especially for students that have never taken a course on mathematical logic—the problems look daunting and confusing, but as more and more problems are solved—and for this it is key, again, to have a huge number of problems to solve— they even become fun to work with. III. CONCLUSIONS The adoption of the OLI platform for the courses Symbolic Logic and Logic and Critical Reasoning with the contents and the design of the Logic & Proofs course has been very successful at the Francisco Marroquín University in terms of the quality of support that the technology provides to teachers and students and the extent for which it is used in the course that ranges from in classroom aid to mid-term exams or final exams. Several key factors have been identified for this success but the list is not exhaustive and others may also be important. IV. FUTURE WORK There are aspects of the case presented here that have not been studied locally or considered for this work. For example, because the course has been taught only with the support of OLI and Logic & Proofs we have no way to compare the performance of students that have not used the platform versus those that have or to compare the effectiveness of teachers that don't use the technology versus those that do use it. We need to devise a way to make that comparison. From communications with researchers at Carnegie Mellon we know that they have carried out experiments comparing the pure on-line version of Logic & Proofs with a lecture based class using
a textbook. A preliminary report can be found in the AProS website: http://www.phil.cmu.edu/projects/apros/. Soon we will be able to study their method and conduct our own experiments similar to theirs. It would be interesting to test the extent of work and success a teacher or a team of teachers, using a standardgeneric platform to develop their own contents, may have in any type of course versus those that adopt a ready-made platform and content. Highly important is also to study how much of the online content is kept in mind by students after the course has ended and if the figure is better for students that have not used on-line courses. In a similar fashion, the depth of comprehension that both types of students achieve may be different. As new courses move to adopt on-line education technology for teaching or supporting their contents, we will have the opportunity to study these topics. REFERENCES 
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AUTHORS Leonel V. Morales Díaz is with the Universidad Francisco Marroquín, 6ta Calle Final, Zona 10, Guatemala, 01010 (e-mail: firstname.lastname@example.org). Moris A. Polanco is with the Universidad Francisco Marroquín, 6ta Calle Final, Zona 10, Guatemala, 01010 (e-mail: email@example.com).