## Electronic Theses and Dissertations

### Calculus Students' Understanding of Logical Implication and its Relationship to their Understanding of Calculus Theorems

Summer 8-21-2015

#### Level of Access Assigned by Author

Campus-Only Thesis

#### Degree Name

Master's of Science in Teaching (MST)

#### Department

Science and Mathematics Education

Natasha Speer

Robert Franzosa

Eric Pandiscio

#### Abstract

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.