#### Date of Award

Summer 8-2016

#### Level of Access

Open-Access Dissertation

#### Degree Name

Master's of Science in Teaching (MST)

#### Department

Science and Mathematics Education

#### Advisor

Natasha Speer

#### Second Committee Member

Billy Jackson

#### Third Committee Member

Eric Pandiscio

#### Abstract

Students’ low success rates in college calculus courses are a factor that leads to high attrition rates from science, technology, engineering and mathematics (STEM) degree programs. To help reach our nation’s goal of one million additional STEM majors in the next decade, we must address the conceptual difficulties of our students. Studies have shown that students have difficulty with the concepts of slope and derivative, especially in cases when students are asked to utilize these concepts in real-life contexts.

For this study, written surveys were collected from 69 differential (first semester) calculus students. Follow-up clinical interviews were performed on 13 integral (second semester) calculus students. Through the surveys and interviews, students’ understanding of slope and derivative using real-life contexts was explored. On the surveys, students answered questions about linear and nonlinear relationships and interpretations of slope and derivative. They also critiqued the reasoning and accuracy of a hypothetical person’s predictions based on values of slope and derivative. In interviews, students explained their thought process and reasoning for the problems, and answered follow-up questions.

Results indicate that students struggle with knowing what the slope and derivative represent and how to use them appropriately to make predictions. The dominant incorrect reasoning by students (one-third of surveyed students and two-thirds of interviewed students) was to think of slope as the ratio-of-totals (y / x) instead of the ratio-of-differences ( Δ y / Δ x) . Thinking of slope as a ratio-of-totals implies that all linear relationships are directly proportional (of the form f(x)=mx, with a y-intercept of zero); students went on to interpret the slope as something that can be used to calculate the value of the dependent variable (by multiplying it by the value of the independent variable).

This incorrect thinking about slope influences students’ understanding of the derivative. As a result, they often interpreted the derivative as something that could be used to find the value of the dependent variable (by multiplying the derivative by the value of the independent variable). This led to the incorrect relationship, f(x) = f'(x) * x . Furthermore, when students were asked to critique the reasoning of a hypothetical person’s predictions, they showed little knowledge of how the derivative can be used to make valid predictions. Instead of demonstrating understanding that the derivative can be used to estimate change only near the input value, 54% of interviewed students said once again that they could use the derivative to calculate the total value (f(x) = f'(x) * x). Students’ impoverished views of slope are adversely impacting their ability to understand the more advanced related topic of derivative.

Knowing more about students’ understanding of slope and derivative as rates of change can help educators improve our instruction, with the overall goal of retaining our STEM majors. Instructional implications of this study, as well as limitations and future avenues for research, are discussed.

#### Recommended Citation

Tyne, Jennifer G., "Calculus Students' Reasoning About Slope and Derivative as Rates of Change" (2016). *Electronic Theses and Dissertations*. 2510.

https://digitalcommons.library.umaine.edu/etd/2510