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Hands-on Mathematics - Algebra, trigonometry, and calculus tailored for engineers help boost retention and student success. + By Mary Lord

It’s orientation season and dozens of University of Vermont freshmen and their families have filed into Votey Hall, home of the College of Engineering and Mathematical Sciences, for some academic survival tips from Joan Marie Rosebush, the college’s director of student success and, not coincidentally, a senior math instructor. “Math is the most important course you’ll take,” she tells the newcomers. “If you’re not solid, you’re just asking for trouble.” Indeed, the inability of incoming freshmen to advance past the traditional introductory calculus sequence – the prerequisite for statics, dynamics, and other core engineering courses – has become a leading cause of attrition and a major challenge for engineering programs nationwide. “It’s worse than a gatekeeper. It’s a bottleneck,” contends Nathan Klingbeil, senior associate dean and professor of mechanical and materials engineering at Wright State University.

We’re trying to propose an engineering solution to the way engineering education works in this country. - Nathan Klingbeil - senior associate dean at Wright State University

Vermont, Wright State, the University of Utah, and Cornell, among other schools, are working to eliminate that bottleneck with math curricula designed for engineering students who arrive at college ill-prepared or rusty. Introducing streamlined precalculus, interactive online summer classes, math with engineering applications, and small-group problem solving guided by teaching assistants, they’ve eased students’ entry to engineering and seen improved retention and graduation rates. In the process, they have pinpointed and sought to build upon important differences in the analytical skills required of mathematicians and engineers.

Summer Refresher

After finding that many freshmen weren’t ready for calculus, Vermont’s Rosebush whittled down a precalculus course, preserving the rigor and textbook but focusing on the math needed for physics. Rosebush also offers incoming engineers an online, no-pressure summer math refresher course, a cross between Khan Academy visuals and small-group tutorial, with step-by-step calculations and student responses. The idea, she says, is to have math “not just for the ones who do belong in engineering, but for those students who think engineering is for them.” The school offers evening drop-in tutoring sessions during the semester, and Rosebush teaches a freshman calculus section that meets five mornings a week instead of the usual four, building confidence as well as competence.

Back in 2003, Wright State was losing most of its aspiring engineers before they completed the required calculus sequence. Today, first-year retention has reached an all-time high, student performance in math and engineering continues to rise, and graduation rates have soared. What changed? Wright State replaced the traditional math prerequisites for core sophomore-level courses with EGR 101, Introductory Mathematics for Engineering Applications, which delivers only the algebra, trigonometry, calculus, and other math topics actually used in physics, circuits, computer programming, and other engineering fundamentals. Developed by Klingbeil and colleague Kuldip Rattan, EGR 101 is taught exclusively by engineering faculty and student TAs, whose lectures, labs, and recitations provide physical context to math. “When you teach math for the sake of math, you develop problem solving and critical thinking skills, but you don’t develop an ability to transition between applications,” says Klingbeil. By letting students move ahead in the curriculum before finishing the required calculus sequence, EGR 101 has pushed engineering graduation rates to 40 percent, compared with 15 percent for those who didn’t take the course.

“The Derivative: What is it, and why do engineers need to know it?” is how EGR 101 introduces calculus. Rather than focus on the equations, Klingbeil has students drop a ball and measure the time to impact. An engineer, he says, would want to know the ball’s average velocity and speed at impact. The first is just distance divided by time. To calculate the latter, however, students must measure the velocity of the ball at different points as it drops, eventually connecting their results to the slope — the definition of the derivative.

To convey Newton’s laws, Klingbeil asks students to calculate the stopping time of a braking car. “There’s not a freshman engineering kid in the country who doesn’t understand that,” he says. Manipulatives also help instill understanding. For example, co-developer Rattan uses one- and two-linked robots to teach trigonometry, which is “the way you actually use trig,” says Klingbeil. Students can take the link, measure it, and plug their results into a formula to see if it works, rather than having to “remember a bunch of trig identities.”

Joshua Deaton is “one of the textbook cases” of a first-year engineering student doomed to derail at Wright State. “I never would have persevered through the calculus,” which his rural Ohio high school didn’t offer. Such EGR 101 engineering examples as figuring the area of asphalt needed to widen a truck entrance — a problem from Klingbeil’s co-op experiences — “saved me,” he contends, and helped switch his mind-set from “I’m doing math to I’m using math to analyze something.” Deaton earned a bachelor’s in mechanical engineering with highest honors in 2009. Now pursuing a Ph.D., he is one of a number of grad student teaching assistants called upon to apply the new freshman pedagogy contained in EGR 101.

When not using complex algorithms to design and model aircraft structures, Deaton is helping freshmen understand differential equations by examining the Tacoma Narrows Bridge disaster. Because he remembers what was “really hard,” he knows which sections his freshmen will stumble over and tailors examples accordingly. Deaton also has written MATLAB programming guides while other TAs have rewritten labs to make difficult topics less daunting. He often calls students to the board and says, “You, come up and solve it.” And no one leaves the room “until everyone gets the problem,” he says. “I’m notorious.”

Incorporating Earth Challenges

Poor student results on an engineering professor’s math placement test at first had University of Utah engineering and mathematics faculty yelling at each other. But now they are revising, together, the first two years of undergraduate engineering math.

The first course, which debuts this fall, will cover single-variable calculus, vector geometry, algebra, and the calculus of parametric curves. It is the beginning of an accelerated four-semester core sequence that formerly took five semesters. Each course will explore engineering applications in TA-led small-group labs, with videos created by every engineering department supplementing lectures and providing the basis for homework problems and class projects. For example, chemical engineering professor and associate chair Geoff Silcox has developed modules on current environmental challenges that involve engineering math, such as the depletion of world oil reserves and the build-up of pollution in lakes. His challenge: Finding examples that didn’t “go well beyond” the students’ engineering background. Utah also is investing heavily to keep class sizes small, train teachers, and intervene with strugglers. “The bottom line is increasing the number of high-quality engineering graduates,” says Peter Trapa, math department chair.

For engineers, “It’s not enough just to know the math,” says Cornell math instructor Maria Shea Terrell, who serves on a multidisciplinary panel working to refine the school’s 50-year-old engineering math offerings. Students might understand a concept, she says, “but that ability to apply it is a separate skill.” To help students figure out how to turn an engineering problem into a mathematical one, Terrell and her colleagues in engineering and math developed materials with an engineering context. A query on surface intervals, for example, might ask the volumetric flow rate of water as it pours through a pipe cut at different angles. As teaching assistants move from table to table, posing questions that steer group discussions, students converge on the correct answer: a bucket would fill at the same rate because the flux across the pipe’s surface never changes, no matter how it is cut. In a traditional math class, students rarely get to see which concept is right for the job.

Engineering math’s applications approach certainly has appeal. Cornell student surveys routinely give the course high marks. Supported by $4.6 million in National Science Foundation grants, Wright State’s model is now under consideration by more than two dozen institutions nationwide, including Oklahoma State University, the University of Tulsa, and the University of Toledo. Whether engineering math takes root beyond a handful of innovators and pilot programs remains to be seen. That certainly is Klingbeil’s aim. “We’re trying to propose an engineering solution to the way engineering education works in this country,” he says. “It needs to be mandatory, and it needs to be wide scale… We’re trying to fix a national problem here.” With the president calling for universities to graduate 10,000 more engineers a year, it doesn’t take a genius to do that math.

Wright State’s model math curriculum is available at

Mary Lord is deputy editor of Prism.


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