Calculus Students\u27 Understanding of Logical Implication and its Relationship to their Understanding of Calculus Theorems

In a typical calculus course, students are often presented with definitions, lemmas, propositions, and theorems. Often, these statements are conditional, that is, they are sentences of the if-then form. For example, the differentiability implies continuity theorem is the following conditional statement: For all functions f, if f is differentiable at x = c, then it is continuous at x = c. Calculus students must be able to make proper inferences from these statements in order to build their calculus knowledge. Calculus is a major aspect of many science, technology, engineering and mathematics (STEM) undergraduate programs. Thus, if an aim of the mathematics education community is to improve the teaching and learning of calculus, it is important that researchers understand how students think about calculus theorems and about logical implication itself. Given the conditional statement “A implies B,” the following four inferences can be made: modus ponens: Suppose A is True. Then B is True. inverse: Suppose A is False. Then it is not known whether B is True or False. converse: Suppose B is True. Then it is not known whether A is True or False. contrapositive: Suppose B is False. Then A is False. Many studies have investigated participants’ abilities to carry out the four tasks above in a variety of contexts including verbal (English language, syllogism-like tasks) and symbolic (abstract tasks). These studies show that, generally, participants of all ages struggle with conditional reasoning. What has not been pursued, however, is how undergraduate calculus students make conditional inferences given a calculus theorem and given an abstract proposition. That is, no studies have examined how undergraduate calculus students’ understanding of logical implication in the abstract relates to their understanding of calculus theorems. Fifty-two calculus students were given a multiple-choice survey consisting of the four tasks above. The instrument had two parts. The first part features the four types of tasks based on a calculus theorem while the second part features the same four tasks for an abstract proposition (e.g. if a ~ b, then aa ~ bb). Ten participants were also interviewed concerning the tasks given in the survey. Results show that participants performed well on the calculus portion of the instrument, but poorly on the abstract portion. Further results suggest that this is due to participants’ abilities to use their calculus knowledge to generate examples. This leads many participants to correctly answer the inverse and converse tasks. Concerning the abstract tasks, many participants engage in truth-value matching. That is, given a true premise, they respond with a true conclusion and, given a false premise, they respond with a false conclusion. This kind of thinking is known as “Child’s Logic” and has been documented in the mathematics education literature as well as the psychology literature. The results also suggest that some students answer these tasks based on some understanding of conditionals. Limitations of the study, as well as teaching and research implications, are also discussed