In many pre-college curricula, informal and formal logic does not exist as its own course, even though thinking logically is crucial to much academic work. As students progress through their primary and secondary educations, they will be expected more and more to support their opinions with well-supported arguments. While schools may find it difficult to fit a logic course into their curriculum, they might find it easier to devote small portions of their existing courses to learning specific logical tools. Through the years, I have frequently taken time in my own history classes, as well as working with my mathematics colleagues, to teach logical thinking. I have often done so, however, by using interactive demonstrations to help students grasp the tools I am teaching. In this blog, I will share four activities that teachers can easily adopt should they wish to add a logic component to their courses.<\/p>\n
Human Venn Diagrams<\/b><\/p>\n
Students are likely to encounter Venn diagrams in their math classes. Since a fair number of students struggle with Venn diagrams, it sometimes helps to show them how to visualize what is happening when they are constructing diagrams for syllogisms, especially when it comes to working out the diagrams for the two premises. An interactive way to do this is to have them play \u201chuman Venn diagrams.\u201d<\/p>\n
Begin by drawing the Venn diagram for a syllogism on the floor of your classroom or in the playground. One can use a variety of methods to represent the circles: from taping or stapling colored strings to chalk to large poster board. To represent shaded out regions in the diagram, use black construction paper that students can put in the appropriate regions. Next, create three categories that apply to your students. For example, \u201cstudents wearing glasses,\u201d \u201cstudents wearing watches,\u201d and \u201cstudents possessing cell phones.\u201d Next, working in groups, students should construct several syllogisms using the categories. For example, one group might formulate the following argument:<\/p>\n
(1) All students wearing watches are students who have cell phones.<\/p>\n
(2) Some students who have cell phones are students who do not wear glasses.<\/p>\n
(3) Therefore, no students who wear glasses are students who wear watches.<\/p>\n
Assign each student a premise. Students then must stand in the correct place on the diagram and shade regions using the construction paper when necessary. Any student standing in a region means that there exist members of that particular category. Students should act out one premise at a time, making adjustments to their positions on the diagram when new information comes to light because of the second premise. Lastly, a student should stand in the region that represents the conclusion on the diagram. Students should conclude the exercise by explaining how their human Venn diagram has established the validity or invalidity of the syllogism.<\/p>\n
I have used this method successfully with middle school and first and second year high school students. Students find that this exercise, while fun, helps them better picture what is going on in the diagrams. I recommend that teachers also access their students\u2019 understanding by having them explain what is going on in their diagram as they act it out. One way to do this is to have each participant be responsible for saying why she is standing where she is, why she was forced to move to a new region, or why she used the black construction paper.<\/p>\n
Fallacy of the Day<\/b><\/p>\n
Many of my students get excited to learn about the informal fallacies, if the cries of \u201cad hominem!<\/i>\u201d or \u201cstraw man!<\/i>\u201d in the halls and classrooms of my school are any indication. Indeed, learning the fallacies is invaluable for developing our students\u2019 ability to evaluate and critique arguments and positions.<\/p>\n
I have often introduced my own students to the informal fallacies by devoting about ten minutes of class time once a week to explaining and discussing a specific fallacy. Over the course of a typical thirty-three week school year, students will learn a fair number of common mistakes in reasoning. I have found that the time spent is often well worth it, as students feel empowered by recognizing these fallacies and will often actively look for them in their readings. Some will, naturally, go to the \u201cdark side,\u201d by using their newfound knowledge intentionally sophistically, but, perhaps unsurprisingly, they find it harder to get away with manipulating other students because their peers have also learned the fallacies. Needless to say, this quite clearly demonstrates to my students the value of knowing the fallacies.<\/p>\n
It helps to reinforce the fallacy of the day by having an example of it in the reading assigned for that same class. For example, consider Pope Urban II\u2019s call for the First Crusade:<\/p>\n
Let the deeds of your ancestors encourage you and incite your minds to manly achievements<\/span>:-<\/span>the greatness of King Charlemagne, and of his son Louis, and of your other monarchs, who have destroyed the kingdoms of the Turks and have extended the sway of Church over lands previously possessed by the pagan. Let the holy sepulcher of our Lord and Saviour, which is possessed by unclean nations, especially arouse you, and the holy places which are now treated, with ignominy and irreverently polluted with the filth of the unclean.<\/span>1<\/sup><\/a><\/span><\/sup><\/p>\n If the fallacy of the day was <\/span>argumentum ad popular, <\/i>one could use this text not only to illustrate the fallacy, but also to discuss why it was so effective.<\/p>\n \u201cWhat the Tortoise Said to Achilles\u201d<\/b><\/p>\n Teachers wishing to add some logic into their classes might find Lewis Carroll\u2019s <\/span>Alice\u2019s<\/i><\/span>Adventures In Wonderland<\/i><\/span> and <\/span>Through the Looking Glass <\/i><\/span>useful. My own students find the stories fun to read and also enjoy discussing their playful use of logic. One approach for using Carroll is to break students into small acting troupes whose job is to perform a scene from the text that illustrates a logic puzzle. After they present the scene, the troupe can then either explain its logical significance or lead the rest of the class in a discussion about it. <\/span><\/p>\n Carroll can also be used to teach students about logical paradoxes. Students often feel personally challenged to resolve the paradoxes of Zeno or Russell, and having a section on them can help students understand the difference between contradiction and paradox. I often try to show my students that a logical contradiction can be resolved by simply giving up on one of the propositions causing the problem. With a paradox, however, I want my students to see that giving up either contradicting proposition is ultimately unsatisfactory. The paradoxes of the \u201cDichotomy\u201d and \u201cAchilles and the Tortoise\u201d are particularly troubling for students as they \u201cknow\u201d that moving is possible, yet Zeno\u2019s reasoning seems correct.<\/span><\/p>\n Students can also formulate a paradox for homework. They can either be asked to mimic Zeno\u2019s argument forms, or \u2013 even better \u2013 try to develop their own paradoxes. Students will find the latter difficult, but their efforts can be used to have instructive discussions on how contradiction and paradox differ from each other. <\/span><\/p>\n \u201cDuck; Duck; Refute!\u201d<\/b><\/p>\n One exercise that helps students practice both public speaking and thinking on their feet is a variant of the old \u201cDuck, Duck, Goose\u201d game. Students stand in a circle with the student speaking first holding a tennis ball. Based on the previous night\u2019s reading, the instructor should have some controversial point about the assignment to start. The first student makes an argument about the point, usually by arguing for or against it. Once finished, she throws or passes the tennis ball to another student who must try to refute the first student’s position. The third student attempts to defend the first student by refuting the second, and so on.<\/p>\n Take, for example, the famous trolley problem where a person needs to decide either to allow five people standing in the path of a runaway trolley to be killed or to pull a switch that changes the trolley to another track where it will kill only one person. For this exercise, the first student would begin by offering an argument, such as \u201cthe person must divert the trolley because five lives are worth more than one.\u201d The student would then throw the tennis ball to another student who would have to raise an objection to the initial argument: for instance, \u201cBut, killing one to save five violates the rights of the one.\u201d This student would then throw to a third student, and so on.<\/p>\n I find this exercise works well for brainstorming about the material, but I have also used it to train students in public speaking and arguing for Model United Nations conferences.Depending on class size, the rest of the students can serve as a jury that can debate and decide which side of the argument has the better position.<\/p>\n To conclude, for schools where informal and formal logic courses are absent, these activities and others like them can help students learn some logical thinking in a fun and minimally intrusive way within an established curriculum. Students will benefit from thinking more clearly and rigorously in many of their courses, especially those where formulating and defending an argued position are central.<\/p>\n