Mathematical Maturity for Engineering Students

Abstract

This dissertation presents four studies on the mathematical education of engineering students. The first study is a qualitative analysis of the beliefs of engineering faculty at a single institution regarding what constitutes "mathematical maturity" for engineering students. Faculty emphasized the need for mathematical modeling skills, fluent symbolic representation skills, and a combination of effortless algebraic fluency and ability to use computational tools. The second study is an analysis of the beliefs of engineering faculty at a variety of institutions. These faculty also emphasized modeling, representation, and computation, corroborating the results of the first study. The third study is an analysis of the mathematical content of engineering circuits and statics homework problems. Just 8% of statics problems and 20% of circuits problems use calculus, and in a much more limited way than what is taught in calculus. The fourth study presents a quantitative survey of engineering sophomores' perceptions of the relevance of mathematics to their engineering studies. The students have somewhat favorable views of the relevance of mathematics, but some high-performing students view mathematics as irrelevant.ii To David Pierce, for starting me on the path to becoming an educator.And to Caitlin McGuire, for helping me complete that journey.iii

First mathematics course in college and graduating in engineering: Dispelling the myth that beginning in higher‐level mathematics courses is always a good thing

“…However, upon entering college, many students are not ready to enroll in a Calculus course as their first mathematics course. Mathematics readiness plays a major role in student retention and degree completion in engineering programs (Faulkner et al, 2019; Gardner et al, 2007; Geisinger & Raman, 2013; Greefrath et al, 2017; Madison et al, 2015; Redmond‐Sanogo et al, 2016; Tyson, 2011). In general, when students are adequately prepared in mathematics in high school, they are more apt to enroll in advanced mathematics courses in college and might earn higher grades in those courses.…”

When am I ever going to use this? An investigation of the calculus content of core engineering courses

“…Skills that are not applied in subsequent courses might also be an obsolete topic as tasks requiring those skills are relegated to computers and no longer in active use in engineering, such as exact equations and integrating factors commonly taught in ordinary differential equations courses . The obsolescence explanation is suggested by interviews with engineering faculty (Cui, 2006;Faulkner et al, 2019;Faulkner, Herman, & Earl, 2017;Ferguson, 2012;Ganter & Barker, 2004;Varsavsky, 1995). Note that this would not apply to concepts as a lack of conceptual understanding has been shown to lead to erroneous diagrams in statics (Johnson-Glauch & Herman, 2019).…”

“…Various lenses such as "mathematical thinking" (Cardella, 2008;, "mathematical epistemology" (Gainsburg, 2015), "mathematical competencies" (Alpers et al, 2013) or "mathematical maturity" (Faulkner et al, 2019;Ferguson, 2012) capture meta-mathematical knowledge and ways of thinking that go beyond specific content or techniques. However, curriculum design choices are often justified in terms of topic knowledge.…”

“…While many confounders likely contribute, a reasonable candidate would be algebra skills, since calculus courses and engineering courses both require large amounts of algebraic calculation. Some engineering instructors explicitly claim that they require calculus as a prerequisite entirely for the algebra practice, even if their course uses no calculus at all (Faulkner et al, 2019).…”

Calculus at the intersection of institutions, disciplines and communities: a special issue Guest Editorial

“…Calculus is also widely recognized as a critical stage in various transition processes, such as the transition from secondary mathematics to tertiary mathematics, the transition within and across university courses, or the transition from university to the workplace (Hochmuth et al, 2021). There is ample evidence that students around the world struggle in their calculus courses, that rates of failure in these courses are comparably high, and that students' experiences in calculus courses are strongly linked to student dropout in STEM programs (Artigue et al, 2007;Faulkner et al, 2019).Considering the critical role and varied goals of calculus in different contexts, it may seem contradictory that calculus is very often taught de-facto as one and uniform…”